August, 1979
ESL-FR-834-9
COMPLEX MATERIALS HANDLING AND ASSEMBLY SYSTEMS
Final Report
June 1, 1976 to July 31, 1978
Volume IX
ANALYSIS OF TRANSFER LINES CONSISTING OF THREE UNRELIABLE
MACHINES AND TWO FINITE STORAGE BUFFERS
by
Stanley B. Gershwin
Irvin C. Schick*
This research was carried out in the M.I.T. Electronic Systems
Laboratory (now called the Laboratory for Information and Decision
Systems) with support extended by National Science Foundation Grants
NSF/RANN APR76-12036 and DAR78-17826.
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Mass. 02139
Presently with Scientific Systems, Inc., Cambridge, MA
ABSTRACT
An important class of systems, which arises in manufacturing, chemical
process, and computer contexts, is one where objects move sequentially from
one work station to another, and where they rest between stations in buffers.
In the manufacturing context, such systems are called transfer lines or
flow shops.
In the research reported here, the dynamic behavior of a buffered transfer line with unreliable work stations is modelled as a Markov chain.
The
system states are defined as the operational conditions of the stages and the
levels of material in the storages.
The steady-state probabilities of these
states are sought in order to establish relationships between system parameters
and performance measures such as production rate
(efficiency), forced-down
times, and expected in-process inventory.
The steady state probabilities are found by choosing a sum-of-productsform solution for a class of states, and deriving the remaining expressions
by using the transition equations.
In this way, the order of the system of
equations to be solved is drastically reduced.
Various properties of this
reduced-order system are discussed, as well as methods to improve its numerical
behavior.
This algorithm suggests a general technique for solving large-scale,
structured Markov chain problems.
ii
PREFACE
In order for this volume to be self-contained, some material in
Chapters 2 and 3 reproduce material that has already appeared in Volume
VI of this report (Schick and
Gershwin [1978]).
Thanks are due to Mr. Wei-Tek Tsai for his early help with the
computer program and Fig. 4.1.
Mr. Steven C. Glassman did an excellent
job in writing the final version of the computer program, and he and
Dr. Alan J. Laub and Ms. Virginia C. Klema provided useful advice on
the singular value decomposition method.
Ms. Klema and Mr. John E. Ward
also contributed editorial comments on the manuscript.
Additional
analytical assistance and experience with the program was provided by
Ms. Brenda Pomerance.
We thank Ms. Margaret Flaherty for typing the manuscript and Mr.
Arthur J. Giordani and Mr. Norman Darling for drafting the figures.
We are most grateful to Dr. Bernard Chern of the National Science
Foundation for his continuing sponsorship of our research in automated
manufacturing and materials handling systems.
During the period June 1, 1976 to July 31, 1978, this research was
supported by National Science Foundation Grant NSF/RANN APR76-12036.
Since August 1, 1978, it has been supported by National Science
Foundation Grant NSF/RANN DAR78-17826.
iii
TABLE OF CONTENTS
Page
ABSTRACT
ii
PREFACE
iii
TABLE OF CONTENTS
iv
LIST OF FIGURES
vi
LIST OF TABLES
vii
1.
1
2.
3.
INTRODUCTION
1.1
Past Research
3
1.2
Overview of the Method
4
1.3
Outline of Report
5
6
MODELLING
2.1
The Unreliable Transfer Line with Interstage
Buffer Storages
6
2.2
State Space
7
2.3
Assumptions of the Model
8
2.4
The Markov Chain Model
9
2.5
General Results from Markov Chain Theory
15
2.6
Performance Measures
17
ANALYTICAL SOLUTION OF TRANSITION EQUATIONS
20
3.1
Internal State Transition Equations
20
3.2
The Sum-of-Products Solution for Internal State
Probabilities
22
3.3
Boundary State Expressions
26
3.4
3.3.1
Transient States
28
3.3.2
Boundary States Reachable from Internal States
in One Step
31
3.3.3
Other Expressions of Internal Form
33
3.3.4
Other Expressions
33
Reduction of the System of Equations
iv
38
Page
4.
5.
6.
CONSTRUCTION OF THE PROBABILITY VECTOR
42
4.1
Analysis of the Parametric Equations
42
4.2
Analysis of the Reduced-Order System of Equations
4.3
Limiting Behavior of U
60
4.4
Limiting Behavior of i(u)
70
DISCUSSION OF METHOD AND RESULTS
5.1
Solution of Reduced-Order System: Memory
Requirements and Numerical Difficulties
5.2
Qualitative Discussions of the Solution
75
5.2.1
Magnitudes of i Expressions
77
5.2.2
Values of Uj, j = 1,..,
79
CONCLUSIONS AND AREAS OF FUTURE RESEARCH
6.1
Different Boundary Expressions
6.2
Choices of Uj, j= 1,...,
6.3
Alternative Models
81
81
82
84
APPENDIX
A.
A SET OF BOUNDARY STATE TRANSITION EQUATIONS
B.
(s.,U) EXPRESSIONS WHICH ARE NON-ZERO AND NOT OF
INTERNAL FORM
C.
LIMITING
D.
k (s) EXPRESSIONS
k
A COMPUTER PROGRAM FOR SOLVING THE THREE-STAGE
TRANSFER LINE PROBLEM, WITH SAMPLE OUTPUT
REFERENCE S
85
97
101
122
122
194
LIST OF FIGURES
2
Figure 1.1: A k-stage transfer line
Figure 4.1:
Locus of
(X1
X 2)
Page
parameters
45
Figure 4.2:
Graph of equation (4.15)
48
Figure 4.3:
Non-zero
k[s]
74
regions
vi
LIST OF TABLES
Table 2.1:
Transition Probabilities for Machine Operating
Conditions
11
Table 2.2:
Storage Level Transitions
12
Table 3.1:
Transient States
29
Table 3.2:
Boundary States Reachable from Internal States in
One Step
32
Table 3.3:
Additional States of Internal Form
34
Table 3.4:
Expressions Obtained in Pairs
36
Table 3.5:
Expressions Obtained Singly
39
Table 4.1:
Limiting Values of Z. and W.
J
J
49
Table 4.2:
Bounds on Sets A,B, and C
50
Table 4.3:
Sign Combinations for the Curves on Figure 4.1
52
Table 4.4:
Q, The Set of Odd States
55
Table 4.5:
Limiting Qj Combinations
66
vii
1.
INTRODUCTIQN
An important class of systems, which arises in manufacturing, chemi-
cal process, and computer contexts-, is one where objects move sequentially
from one station to another, and where they rest between stations in buffers.
In the manufacturing context, such systems are called transfer lines, and
the stations are transfer machines.
A schematic diagram of such a system
appears in Figure 1.1,
In the research reported here, the dynamic behavior of a transfer line
is modeled as a Markov chain, and a method is proposed for finding the
The
steady state probability distribution of the states of that chain.
probability distribution is used to calculate such measures of performance as average efficiency
inventory.
(production rate) and average in-process
The method is applied here to three-stage systems.
simpler form, the method has been applied to two-stage systems
and Gershwin
[1978], Gershwin and Berman
[1978], Berman
In a
(Schick
119791).
It
is hoped that the method presented here can be extended to longer lines
and more complex networks.
In the model discussed in this report, the source of randomness is
the unreliability of the workstations or machines.
The machines fail
at random times and remain inoperable for random periods during which
they are under repair.
It is possible to compensate for workstation
failures by providing redundancy, i.e., secondary parallel stations that
are brought into use in case of failures of primary machines.
This
process, however, can be prohibitively expensive if system components
are costly.
An alternative consists in placing buffer storages between unreliable
stages.
These provide temporary storage space for the products of
stations upstream of a failed station.
Similarly, they provide temporary
supplies of workpieces for stations downstream of a failed station.
Thus, they act so as to decrease the effects of workstation failures
on the rest of the line.
However, costs of floor space, material handling equipment, and
in-process inventory are also of great importance.
-1-
It is thus necessary
-2-
V
-
0
F*0)
o
L
(D
0
9
+
*-
44
-
l-
Io
U
CV)~~~~~()
0)
rjC
L
m
Cm
m
X
C
H
~~H
*o
0
-
~~~~~~~~~~~~~~~~~-x.
0
H)
0)
V)
-
10a
C
~
-3-
to find in some predefined sense the "best" storage configuration;
leads to an optimization problem.
this
To solve this problem, it is essential
to be able to quantify the relations between transfer line design parameters (i.e. average up and down times of workstations, storage capacities)
and the performance measures.
1.1
Past Research
Transfer lines were first (Buzacott [1967a])
through a probabilistic
studied analytically
approach by Vladzievskii [11952].
Applications
of queueing networks and transfer line models are found in a wide range
of areas, including computer science, coal mining, the cotton, paper
and chemical industries, aircraft engine overhauling,
and the automotive
and metal cutting industries; an extensive literature survey of related
work appears in Schick and Gershwin [1978].
The production rate of transfer lines in the absence of buffers
and in the presence of buffers of infinite capacity had been studied
by many researchers, including Buzacott
[1967a, 1968], Hunt [1956],
Suzuki [1964], Rao [1975a], Avi-Itzhak and Yadin [1965], Morse [1965],
and Barlow and Proschan [1975].
Some authors analyze reliable transfer
lines where buffers are used to reduce the effects of fluctuations in
non-deterministic service times (Neuts [1968, 1970], Muth [1973],
Knott [1970a, 1970b], Hillier and Boling [1966], Hatcher [1969],
Patterson [1964]).
Unreliable two-stage systems with finite buffers have been studied
(Artamonov [1976], Gershwin [1973a], Gershwin and Schick [1977],
Gershwin and Berman [1978], Berman [1979], Buzacott [1967a, 1967b,
1969, 1972], Okamura and Yamashina [1977], Rao [1975a,
Sevast'yanov [1962]).
1975b],
Longer systems are more difficult to analyze
because of the complexity of machine interference when buffers are full
or empty (Okamura and Yamashina [1977]).
Such systems have been formulated in many ways (Gershwin and
Schick [1977], Sheskin [1974, 1976], Hildebrand [1968], Hatcher [1969],
Knott [1970a, 1970b],
Buzacott and Hanifin [1978]), and studied by
-4-
approximation
(Buzacott [1967a, 1967b],
Sevast'yanov [1962], Masso [1973],
Masso and Smith [1974]), as well as simulation (Anderson [1968],
Anderson and Moodie
[1969], Hanifin, Liberty and Taraman [1975], Hanifin,
Buzacott and Taraman [1975], Barten
[1962], Freeman [1964], Kay [1972],
Ho, Eyler and Chien [1979]), but no analytic technique has been found
to obtain the expected production rate of a multistage transfer line with
unreliable components and finite interstage buffer storages.
and Gershwin
Schick
[1978] propose numerical, as well as analytical methods
for solving this problem.
This report completes the analytical solution
proposed there.
1.2
Overview of the Method
To find the steady-state probability distribution of a Markov chain,
it is necessary to solve a set of M linear transition equations in M
unknowns, where M is the number of states of the chain.
In the problem
discussed here, M is large, so an efficient method is required.
This problem does have a structure that can be exploited.
Due
to that structure, it is possible to find Z vectors _j (j = 1,...,
each of which satisfies M-k of the transition equations.
Cj vectors fails to satisfy the same £ equations.
i),
Each of the
Consequently if the
probability vector is expressed as a linear combination of these vectors
p =
C'
(1.1)
j=l '
then it is guaranteed to satisfy the M-2
equations each
.j
satisfies.
In order to satisfy the remaining equations, the coefficients Cj must
be appropriately chosen.
To do this requires solving R linear equations in Q unknowns.
Since
Z is much smaller than M, this is relatively easy to do.
The method is not without limitations.
the k unknowns C 1,...
,
CA are poorly behaved.
to use extended precision
First, the k equations in
It has been necessary
(32 decimal place) arithmetic to obtain 5
decimal place precision in analyzing transfer lines with large storages.
Second, 2 increases with the storage sizes.
Even though 2 increases
more slowly than M, the number of system states, it still limits the
size of problem that can be treated.
Also, this increase prevents the
method, as currently formulated, from being usefully applied to longer
lines.
1.3
Effort is being devoted to overcoming these limitations.
Outline of Report
The problem is formally presented in Chapter 2.
is described and the state space is formulated.
The transfer line
The modelling assumptions
are stated, and a Markov chain model is introduced.
Formulas for the
calculation of performance measures are derived.
Steady state transition equations are solved analytically in
Chapter 3.
The equations involving only internal states are satisified
by a sum-of-products form for the steady-state probabilities.
Expres-
sions are also obtained for the steady-state probabilities of boundary
states.
The system of equations is then reduced in dimension.
The steady-state probability vector is derived in Chapter 4.
The
sets of solutions of the four parametric equations in five unknowns are
analyzed and an efficient algorithm for computing solution points is
introduced.
The reduced-order system of equations is studied and its
solution is discussed.
The limiting behavior of probability expres-
sions as solutions of parametric equations approach limits is investigated, towards rendering the system of equations better conditioned.
The reduced-order system of equations is solved, and shortcomings
of the algorithm, (memory requirements, numerical problems) are discussed
in Chapter 5.
A qualitative discussion of the solution is given, and in
Chapter 6, tentative conclusionas pertaining to the method are presented.
2.
MODELLING
A formal statement of the problem is given in this chapter.
A multi-
stage transfer line with unreliable components and interstage buffer
storages is described in Section 2.1 and a state space formulation is
introduced in Section 2.2.
The assumptions made in formulating the mathematical model are discussed in Section 2.3.
Most assumptions are standard.
(See Buzacott and
Shanthikumar [1978].)
A Markov chain model is introduced in Section 2.4 and the properties
of such systems which are applicable to the present problem are discussed
in Section 2.5.
The performance measures are expressed as functions of
state probabilities in Section 2.6.
The Unreliable Transfer Line with Interstage Buffer Storages
2.1
The system under study consists of a linear network of servicing
stations (machines) separated by finite capacity buffer storages (Figure
1.1).
Workpieces enter the first machine from outside the system.
Each
worpiece is processed by machine 1, after which it moves to storage 1.
The part moves in the downstream direction, from machine i to storage
i and on to machine i+l, until it is processed by the last station,
machine k, and leaves the system.
The specific nature of the machine operations is of no consequence
in the present analysis.
In a metal working line, it may consist of
drilling or welding; in a computer network, the operation may be data
processing by a specific computing, storage, or input/output unit.
is assumed, however, that the machines are synchronized.
It
That is,
there is a common cycle time, and all machines that are operating on
pieces start at the same instant.
The buffer is a storage element.
Parts pass through a buffer with
a transportation delay which is negligible compared to service times
in the machines, except for the delay caused by other parts in the queue.
Machines fail occasionally.
Failures may have many causes and thus,
the down-times of failed machines, like the up-times of operating
machines, are random variables.
When a failure occurs, the level in the
adjacent upstream storage tends to rise.
-6-
If the failure persists long
-7-
enough, that storage fills up and forces the machine upstream of it to
stop processing parts.
blocked.
Such a forced down machine is referred to as
Similarly, the level of the adjacent downstream storage tends
to fall during a failure, as the downstream machines continue to drain
its contents.
If the failure persists long enough, the adjacent down-
stream storage empties and the machine downstream of it stops processing
parts.
Such a forced down machine is referred to as starved.
These
effects propagate up and down the line if the repair is not made promptly.
By supplying workpieces, or room
for workpieces to be put in, inter-
stage buffer storages partially decouple adjacent machines.
While machine
failures are to some extent inevitable, the effects of a failure of one
of the machines on the operation of others is mitigated by the buffer
storages.
When storages are empty or full, however, this decoupling
effect cannot take place.
Thus, as the capacities of storages increase,
the probability of storages being empty or full decreases and the effects
of failures on the production rate of the system are reduced.
An inevitable consequence of buffers is in-process inventory.
As
buffer capacities increase, more partially completed material is present
between processing stages.
The effects of interstage buffer storages on the transfer line production rate and on the average in-process inventory are studied here
by formulating a state space description of the system and obtaining
the steady-state probability distribution,
2.2
State Space
It is natural to formulate the problem
of calculating transfer line
performance parameters as one of analyzing a Markov chain.
This is
because the probability of finding storages at a given level or machines
operational or under repair after a cycle depends on the storage
and machine conditions before that cycle.
level
In the Markov chain considered
here, the state of the system consists of the number of workpieces in
each storage and the operating conditions of each machine.
For each machine in a k-stage t'ransfer line, its operating condition
.
is defined by
0
if machine i is under repair,
a. =
1
i = 1,..., k
~1
(2.1)
if machine i is operational,
Here, operational means that the machine is capable of processing a
workpiece.
Whether or not it is actually processing a piece depends on two
additional factors.
At least one part must be present in the storage
upstream, and at least one empty slot must exist in the storage downstream.
(Certain authors, e.g., Kraemer and Love [1970] , Okamura and Yamashima
[1977], define two additional machine states, for times when a machine is
starved or blocked.
However, this is not necessary.)
For each storage j, the variable nj is defined to be the number of
workpieces in the storage.
Each storage has a maximum capacity N., i.e.,
0 < n. < N.
j =
1,...,
k-1 .
(2.2)
The state of the system at time t is defined as
s(t) = (n
)
.
nk-l ( t ) , al( t ), .--,
.
k( t) )
(2.3)
where t, an integer, denotes time in machine cycles.
From equations (2.1) and (2.2), it follows that the number of all
possible system states is given by
M = 2k(Nl+ 1)... (N
+
1) .
(2.4)
In order to calculate such quantities as the average production rate,
the average quantity of in-process inventory, and the fraction of time
each storage is full or empty, the probability of each of the M states
of equation (2.3) must be calculated.
Since M is a large number, an
efficient way to calculate these probabiliites is discussed here.
2.3
Assumptions of the Model
The following assumptions are made, in order to render the mathematical
model tractable while not losing
sight of the physical properties of the
system:
(i) An inexhaustible supply of workpieces is available upstream
of the first machine in the line, and an unlimited storage area is present
-9-
downstream of the last machine.
Thus, the first machine is never starved,
and the last machine is never blocked.
(ii)
All machines operate synchronously with equal deterministic
service times.(This assumption should be compared with those of Gershwin
and Berman [1978]
and Berman [1979].)
cycle takes one time unit.
Time is scaled so that a machine
Transportation takes negligible time compared
to machining times.
(iii)
Machines are assumed to have geometrically distributed times
between failures and times to repair.
If machine i is processing a work-
piece, there is a constant probability pi of machine i failing (i.e., of
ai going from 1 to 0).
mean operating time
This probability equals the reciprocal of the
(in cycles) between failures.
Similarly, there is
a constant probability ri of repair (i.e., of ai going from 0 to 1),
given that machine i has failed (that a. = 0).
of the mean time
(iv)
This equals the reciprocal
(in cycles) to repair.
Machines only fail while processing workpieces.
machine i is operational
Thus, if
(ai = 1) and starved (ni-1 = 0) or blocked
(ni = Ni), it cannot fail.
(v)
line.
Workpieces are not destroyed or rejected at any stage in the
Partly processed workpieces are not added into the line.
When a
machine breaks down, the workpiece it was operating on is returned to
the upstream storage to wait for the machine to be repaired so that
processing can resume.
(vi)
The convention adopted is that a cycle begins with a transi-
tion in the machine operating conditions and ends with a transition in
storage level.
The
The latter is determined by the new machine states.
probabilistic
model of the system is studied in steady state.
Thus, all effects of start-up transients have vanished and the system
may be represented by a stationary probability distribution.
2.4
The Markov Chain Model
By assumption (iii)
of Section 2.3, a machine that is processing
a part has a probability of failure Pi,
When the machine is operational
-10-
but forced down (either starved or blocked), it cannot fail.
Thus, the
failure probability of a starved or blocked machine is zero.
When
processing a part, a machine can either fail or successfully complete
the machining cycle; since its failure probability is Pi, the probability
that an operating machine remains operational is 1-p..
The probability that a failed machine is repaired by the end of any
cycle is r..
This probability is independent of storage levels.
A failed
machine remains down at the end of a cycle with probability 1- r..
These
transition probabilities are summarized in Table 2.1.
Once machine transitions take place, the new storage level is determined
This value is dependent on the new states of the adjacent
(Assumption vi).
machines.
If
the upstream machine is processing a part, the part is added
to the storage; if the downstream machine is processing a part, it is reThe new storage level also depends on the storage
moved from the storage.
levels immediately upstream and downstream at the end of the previous
cycle.
For example, if machine i is operational (ai(t+l) = 1) and the
upstream storage was not empty (ni
i.
(t) > 0), a new piece enters storage
Storage level transitions are listed in Table 2.2.
These probabilities are used in obtaining the state transition
probabilities, defined by
(2.5)
T(i,j) = probIs(t+l) = ils(t) = j] .
This matrix is related to the probabilities in Tables 2.1 and 2.2
by
k(t+l))I
,+ l )
prob[s(t+l) = (nl (t+l),..., nk-ll(+l),...,
(t
==n ((t)
s
()n
l(t)...
nkl(t)
k
1(.
]
k-l
probln
i(t+l)
Ini_lt)
a
(t+l),
ni (t),
ai+l(t)
,
i+l)I
i=l
k
I
prob [i(t+l)lni_l(t),
ai(t), ni(t)]
i=l
(2.6)
prob[ai ( t + l ) Ini
ni-l()n.(t)
, ai ( t ) , n i ( t )
]
a(t+l)
.i
probability
o
0
1-r.
1
10
r.
0
1
1
1
N.
1
0
0
N.
1
1
1
n.(t)
~O
o
l( t )
-1~0
Ni
o0
1
> 0
< N
1
0
Pi
> 0
< Ni
1
1
-Pi
Table 2.1:
Transition Probabilities for machine
operating conditions.
-12TABLE 2.2
prob[ni
ni-l
i-i
(t
(t+l
n.(t)
1
o0
o
)
ni
(t),
l
ni+l (t)
(t)
<
i+l.
Ni+l
0
*
o
>0,<N,
1
< N
i+l
Storage Level Transitions
ai (t+l), n(t), ai+l(t+l),
a(t+l)
a.
(t+l)
i+l
1
0
0
O
0
1
O
1
0
O
1
1
O
O
O
O
~1
0
0
1
1
O
0
0
ni(t)
O
1
n(t)-1
1
o
<N.
i+l
ni (t)
1
n i (t)-1
0
n i(t)
O
1
n i (t)
1
°0
ni(t)
1
1
n i (t)
0
0
N.
0
1
N.-1
1
0
N.
Ni+l0
N.
i
n i (t+l)
0
1
°>0,<N.
ni+(t)]
1
1
N1
0
N.
1 Ni+l
NN
0
0
N
0
1
N.
1
1
0
N.
1
1
N.
1i
1
=
1 if
-13(Table 2.2 continued)
ni-l(t)
>0
>0
>0
>0
>0
> O
>0
(t)
n +(t)
0
i+l
<Ni+
1
0
>0,<N.
1
>0,<N
Ni+
<N
i+l
<N
Ni
<Ni+l
Ni
0
0
N
i+l
il
ni(t+l)
0
1
0
0
1
0
1
1
1
1
0
0
0
0
1
0
1
0
1
1
1
1
0
0
n.(t)
0
1
n.(t) -1
1
0
n i (t)+1
1
1
n.(t)
0
n i (t)
0
1
n. (t)
1
0
n i (t)+l
1
1
n i (t)+1
0
0
0
1
N.-1
1
0
N.
1
1
N.-1
1
0
0
N.
Ni0
Ni+l
N.
+ (
t)(t+l)
1
1
N.
Ni
1
0
1
Ni
1
0
N.
1
1
Ni
1
-14-
where for convenience, no(t)
> 0, nk(t)
<
- and Nk =
,
so that the
conditions no(t) > 0 and nk(t) < Nk are always satisfied.
Since all entries in Tables 2.1 and 2.2 are positive or zero,
(2.7)
T(i,j) > 0 .
It is also possible to show that
T(i,j) = 1.
L
i
To do that,
(2.8)
(2.6) is summed over all values of nl(t+l),..., nkl(t+l),
al(t+l),..., ak(t+l).
1
i:E
this may be simplified to
k
1
In Iprobai(t+l)~E
..
a1(t+l)=0
(t)
i
(t)
n(t)
i
ak (t+l)=0 i=l
k
1
(2.9)
because, once s(t)
of nl(t+l),...,
and al(t+l),..., ak(t+l) are chosen, exactly one set
nkl(t+l) exists which contributes a non zero factor in
That factor is unity.
equation (2.6).
Expression (2.9) can be written
k
1
n
prob
E
i=l 5.(t+l)=
[(t+l) Inil(t), ai(t), ni(t) ]
(2.10)
0
(t+l),
since each factor in the product in (2.9) depends on a single ai.
and since the summations are over all possible values of
(al(t+l),...,
(A proof of a similar exchange of summation and product can
ak(t+l)).
be found in Lemma 3.1 in Section 3.2.)
Expression (2.10) is equal to 1 because, for each i, Table 2.1 indicates that
prob[a (t+l) = 0lni
+ prob[a i (t+l)
This proves
(2.8).
=
ini_
l(t),
l (t)
ai(t), n i (t)]
a.(t), ni(t)]
A matrix that satisfies (2.7),
1
(2.8) is called a
stochastic matrix, and this is required for the model to be a Markov
chain.
(2.11)
-15-
Note that Table 2.2 requires that, with probability 1,
Ini
t + l)
- ni
(t )
(2.12)
I < 1.
That is, in any cycle, a storage gains or loses no more than one piece.
2.5
General Results from Markov Chain Theory
At time t, the probability that the system is in state i is represented
by a state probability vector, p(t), whose components are
p(i,t) = probls(t) =
ii.
(2.13)
They satisfy
p(i,t) = 1,
L
all t
(2.14)
all i
Then, for a Markov chain, the state probability vector at time t+l is
given by
p(t+l) = T p(t).
(2.15)
Recursive application of equation
(2.15) gives
(2.16)
p(t) = T p(O)
where p(O) is the initial probability distribution.
A Markov chain is termed ergodic if the limit
lim T
t
= I
(2.17)
t-*oo
exists and if the steady-state probability vector defined as
p =
p-(O)
'
(2.18)
(whose components are p(i))
is independent of the value of p(O) (Feller
I1966]). As t
(2.15) becomes, for an ergodic chain,
+
a, equation
p = Tp
(2.19)
or
p(i) =
T(i,j) p(j),
all j
all i
-16-
since both p(t) and p(t+l) converge to p.
In the rest of this report, only the steady state probability distribution p is considered.
The following theorems show that equations
(2.14) and (2.19) uniquely determine the value of the steady-state
probability vector.
A closed class is defined as a set of states C such that no state
outside C-can be reached from any state inside C.
Theorem 2.1:
If in a matrix T, all rows and all columns corresponding to
states outside the closed class C are deleted, there remains a stochastic
matrix that defines a Markov chain on C.
This subchain may be studied
independently-of all other states (Feller [1966]).
A process is periodic if a state can be reached from itself in
d, 2d,..., nd steps.
If d = 1 only, the process is termed aperiodic.
States that can always be reached in a finite number of steps after
they are left are termed recurrent.
Theorem 2.2:
Otherwise, they are transient.
In a finite recurrent aperiodic closed class C, the
steady-state probability distribution p is uniquely determined by the
set of equations
/
p(i) = 1
(2.20)
itc
p(i) =
L
T(i,j)p(j)
(2.21)
jec
'(Karlin [1968]).
A final class is a closed class that includes no transient states.
It
may be demonstrated, by showing that any recurrent system state may
be reached from any other recurrent state, that the Markov chain that models
the unreliable transfer line under study contains only one final class.
Furthermore, the existence of a self-loop (a transition such that
T(i,i)
/
0 for some i) on at least one state in a final class is suf-
ficient for its aperiodicity.
There are numerous self-loops in the
-17-
Markov chain under study.
For example, if all machines are operational,
and no storages are full or empty, the system remains in that state
(i.e.,
the same storage levels and all machines operational) with probability
(1- p
)
(1 -
2) ...(1 - pk ) .
For finite storage capacities, equation
(2.4)
indicates that the number of system states M is finite.
It may thus be concluded that equations
(2.14) and (2.19) uniquely
determine the steady-state probability vector for the system under study.
Note that (2.14) and
(2.19) are essentially the same as (2.20) and (2.21)
since the only states i that are not in C are transient, i.e., have
p(i) = 0.
The steady-state probabilities of the transfer line are de-
fined, in accordance with equation (2.3), as
p(s(t)) = p(nl(t),...,
(t)
(t)(t))
kn
.
(2.22)
For reasons presented in Chapter 3, it is necessary to make a distinction between two types of states.
The set of boundary states contains
all states in which at least one of the storages obeys one of the following two relations:
ni < 1
or
(2.23)
at least one i
n. > N. -1
(2.24)
1 -- 1
The set of internal states contains all other states, i.e., all
states for which the relation
2 < ni < Ni- 2
i
1,..., k-l
(2.25)
holds for every storage.
2.6
Performance Measures
The material presented in Chapters 3- 5 is directed towards ob-
taining analytically the steady-state probabilities of all system states.
These probabilities are used in computing important performance measures,
namely efficiency (production rate), in-process inventory, and forced
down probabilities.
Definitions of the performance measures in terms of state probabilities
It is important to note that they are all
are given in this section.
linear functions of the probability vector.
The efficiency E k of a transfer line may be defined as the probability
Since the
that a piece emerges from the line during any given cycle.
machine cycles are fixed and equal (Section 2.3), efficiency is equal to
It is given by (Schick and Gershwin
the production rate per machine cycle.
[1978])
Ek = prob[ak = 1,nkl > 0]
(2.26)
N
1
ik
N
1
k-l
.
..
VE
p(ni'...'n
al
-l'
P(nl,...,nkkl,
n
·
-·
a
l...I
51
..
1)
k-1
1
The forced down times are those events when an operational machine is
unable to process workpieces, either because it has no pieces to process,
or because it has no storage space to dispose of processed pieces.
The
former event, that machine i is starved, has probability given by
PS
= prob[ni_l = 0]
(2.27)
1
11
a l= °
k =0
k
N
n =
N.
Ni
0 ni
NkN.
0
=
1
i-2
P(nl*
t~ni-2' °' ni'
=n
k-l
'
nk-l'
1'
k
Similarly, the event that machines i is blocked has probability
PB. = prob[n. = N.]
B.
1
(2.28)
1
~1N1 1
='S...E
C1=0°
k=0
k
l
Ni-1
Ni
E--- E
n
1= ni_1-
p(nil ...
k-1
Ej *ni+l =
I n i-11 Nil n,+,....
k-l
. n kl'
all ....
a k)
-19Note that at steady-state,
prob[ni-
= 0, ai = 11
= probnin
1
= 0
(2.29)
and
probEn. = Ni,
ai = 1]
= probini = Ni].
(2.30)
This is because a starved or blocked machine can never fail- (by Assumption
iv of Section 2.3) so that states where a failed machine is preceeded by
an empty storage or followed by a full storage are transient
Section 3.3.1).
(Rule i of
Thus, the joint probabilities in equations (2.29) and
and PB reduce to equations (2.27) and
1
Finally, the average number of parts in storage i is
(2.30) which define PS
1
1
ni
E
a1=0
E
all s
N1
Nk-l
2
· * *
*
ak=0
inp(s).
nl=0
(2.28).
n p(nl,.
..
,nk-l'
atl' *
tk)
nk-l=0
(2.31)
3.
ANALYTIC SOLUTION OF TRANSITION EQUATIONS
Transition equations involving internal states only are discussed in
In Section 3.2, a product form is proposed as a tentative
Section 3.1.
solution, and conditions on undefined parameters are found.
This analysis
of internal states and transition equations applies to a general unreliable transfer line of any number of stages.
Boundary state transition equations are introduced in Section 3.3.
These are used to complete the analytical solution specifically of a
three-stage line; the derivation of boundary state probability expressions
for the three-stage line is discussed.
The dimension of the system of
equations is reduced in Section 3.4.
The expressions thus obtained are used in Chapter 4 to compute the
steady-stage probabilities of a three-stage line.
3.1
Internal State Transition Equations
Internal state transition equations are defined as those transition
equations involving only internal states, i.e. equations in which the
final state i as well as all the initial states from which there is
a non-zero transition probability in
equation (2.19) are internal.
When all storages are internal, i.e., when they all have levels
such that
2 < n. < N. -2
-
1
-
i = 1,...,
k -1
,
(3.1)
1
all the operational machines can transfer parts from their upstream to
their downstram storages.
In other words, they are neither starved nor
blocked, and thus remove a piece from the upstream storage and add one
Then, the final level of storage i is given
to the downstream storage.
in terms of its initial
level and the final operating conditions of
adjacent machines by the equation
ni(t+l) = n.(t) +
i(t+l) - c i+(t+l)
This is in keeping with the convention (Assumption
-20-
(3.2)
(vi) of Section 2.3)
-21-
that in each cycle, first machine states and then storage levels change.
Equation (3.2) exactly summarizes Table 2.2 when ni-l(t), ni(t), ni(t+l),
and ni+ (t) are internal.
For internal state transitions, the machine transition probabilities
in Table 2.1 may all be combined in a single expression as
(t), ai(t), ni(t)] =
prob. a(t+l) In
I-a (t1
l-ac (t+l) a.(t+l
il
(l-r.)
E
l-pi~a(t+l)
(3.3)
ai(t)
i1 -i-(t+l)
1-r)
-a. (t+l)
i=l
a. i(t+l) l-a.(t)
.(3.4)
1- i) a i (t+l)p
Pi]1-
Set S(s(t+l))
i
(t+l)ai(t)
is defined to be the set of all states s(t) such that
given ni(t+l), i = 1,..., k-l, and ai(t+l), i=l,..., k, the initial
satisfy equation (3.2).
storage levels ni(t), i = 1,..., k-l,
Equation
(2.19) becomes
p(s(t+l))
=
p(s(t))
T(s(t+l),s(t))
(
s(t)S (s(t+l))
T1(
s
k
n1F
k(t)=O
i=l
Li(t+l),
a
ai(t+l' =li
-(ri
al(t)=o
lri)
pl-Pi
a.(t+1
(t+l)
-(ai 1 -_
Pi
p(nl(t),..., nkl(t),
ota (t)
]
l(t),...,
k(
t ))
(3.5)
-22-
where nl(t),..., nk-l(t) satisfy equation (3.2), i.e. are determined by
s(t+l).
3.2
The Sum-of-Products Solution for Internal State Probabilities
Many queueing theory problems result in product-form solutions.
These have been studied by Jackson [1963]; Gordon and Newell [1967a]
obtained product-form solutions for closed queueing systems with negative
exponentially distributed service times; Baskett, Chandy, Muntz and
Palacios [1975] formulated a theorem applicable to certain types of networks of queues with different classes of customers, stating that the
equilibrium state probabilities are given by a product of factors each
of which is dependent only on one state variable.
Such product form
solutions have also been found by other researchers, including Denning
and Buzen [1977], Lam [1977], Solberg 11977], and Schweitzer [1977].
The work of these authors is concerned with flow through networks of infinite queues, and does not deal with reliability.
It is assumed here that the steady-state probability distribution
for internal states has a sum-of-products form:
~'
P(nl''nkl
p(n. n1
'l''k)
k)
C
j=l jC
E
nl X
Xj
1
k-l
Ylj .
where Cj, Xij, and Yij are parameters to be determined.
ak
Ykj
(3.6)
(3.6)
These parameters
must be such that the internal transition equations (3.5) are satisifed.
In addition, the sum of all probabilities, internal and boundary, equals unity,
E
p(s) =
all s
1.
(3.7)
To satisfy (3.5), these equations are treated as if they were an
ordinary differential equation boundary-value problem.
equation of order n, there may be n distinct solutions.
In a differential
Although each of
these solutions by itself satisfies the equation, only a certain linear
combination of these solutions satisfies the boundary equations (see
for example Boyce and DiPrima [1969]).
Thus, although the true probabilities
-23-
are given by summations of the form of equation (3.6), X.. and Y.. are
13
1J
chosen so that each element in the sum by itself satisfies the internal
transition equations.
If a single term is the summation of (3.6) is substituted into
(3.5), the following is obtained:
CX
n 1 (t+l)
l'
n·
~
''
al(t)=O
J'
a(t
Yk
1
k
1
1
lE-
1)
nk_ (t+l)
' 'Xk-l
ak(t)=O
ca.i(t+l)
[
(l -
) (t + l
ri
(t+l
1-i (t)
i=l
i1- i(t+l)] a i (t)
C
n1 (t)
nk_ (t)
CX1
...X k-l(
t)
(t)
(3.8)
where the subscript j is suppressed for clarity.
Using equation (3.2)
and si.mplifying,
a1 (t+l)
akl 1(t+l)-ak(t+l)
...X'k-1
' 'l Y1
1 (t+l)-a2 (t+l)
X1
a
1
M(t)=O
a.(t+l)
1l-Pi)
1-a (t+l)
Pi
1
Note that ni no longer appears in (3.9).
by
Hk
a. (t+l)
1-a. (t+l)
r.
i=l
leads to
(t+l)
Yk
a.(t+l))
i=l
(k (t)=O
k
r(
[ -(t+l)
1
' ''··C
k
Y.
a.(t)
(3.9)
Dividing both sides of (3.9)
-24a ittl)
ai (t+l)-ai+(t+l)
i-il
k
aLt+l) -
1-a (t+l)
=
)r
a.(t+l)
F-P
1-a. (t+l)
2
°
a0
k
(3.10)
(3t10)
a.(t+l)
r
U-r )
1=
i( t )
Y
1(t+)
-a.
1t)
_
where for convenience, Xk is defined to be 1.
Note that ai(t) only
occurs as an exponent in the right hand side of equations (3.10); furtherThe following lemma is used
more, a. (t) only takes the values 0 and 1.
in simplifying equation (3.10):
For all sets of real numbers {A1 ,..., A k} ,
Lemma 3.1:
A.=
ek=0
a1=0
a,
proof:
k
i
1
(3.11)
(3.11)
(l+Ai)
i=l
i~l
i~l
Proceeding by induction, it is easy to see that equation
(3.11) is satisfied for k=l.
Assuming that the equality holds for k,
the left side of (3.11) is, for k+l,
1
1
1
l
.
(3.12)
:i.
k+l
ak=0
e=°
=1
k+!
1
Ak
1
1
A. 1
='''.Ai
A
k
(1 + Ai )
=
i=1
(by the induction hypothesis)
(
1
+ Ak+l
(3.13)
-25-
(1+ Ai)
(3.14)
Equation (3.14) completes the proof.
Using Lemma 3.1, the right hand side of (3.10) is rewritten as
ai(t+l)
(lPi
)
(1-lr)
l-a i (t+l)
i
r 1
When (3.15) is substituted into (3.10), the argument t (though not t+l)
vanishes.
Simplifying and supressing the t+l argument leads to
k
5
Y.
Xi
ai -
i+l yai
i=1 1
f j[(1 -ri)
a1-a
a. ie
P
a, 1
+ ( 1-P .)
1 r.
Y
.
(3.16)
Equation (3.16) has been derived with no condition on a.; thus, it must
hold for all values of a..
In particular, if ai = 0 for i = l,...,k,
then (3.16) reduces to:
k.
(3.17)
l-ri + piYi]
1 'J
i1l
IT qj 2
1, and
a-
ofor iij,
i = 1,...,k,
then (3.16) becomes
k
X-lXYj =n
[11-r
i
[r. + (-PiYi
.r)Y
(3.18)
{=l
iEj
where, again for convenience, X0 is defined to be 1.
equation (3.18) can be reduced to
Using (3.17),
-26-
X.Y.
r. + (l-p.)Y..
Xj-
1-rj + pDY.
+ p.Y
j=1,...,k .
,l-r
.
Any other sets of values for ai
in equation
may readily be derived from (3.17) and
(3.19) are referred
(3.19)
(3.16) give equations that
(3.19).
Equation
(3.17) and
to as parametric equations in the sequel.
Since X0 = X k = 1, there are k+l equations in 2k-1 unknowns.
Furthermore, the weighting and normalizing constants C. remain to be
computed.
In the three-machine case (k=3), there are four parametric equations
in the five unknowns X 1, X 2, Y1 , Y2 and Y3.
Furthermore, since the
equations are non-linear, there is the possibility of multiple solutions.
Thus, additional information is needed to obtain the steady-state probabilities.
The following notation is introduced:
3.3
U
= (X1-...A Xk-l' Y1'''
Uj
= (Xlj,.''
Yk)
Xk-l,'
(3.20)
j
(3.21)
Boundary State Expressions
In sections 3.1 and 3.2,
n1
(s,U) =
nk
Xk-1
it is shown that expressions of the form
a1
a
... k
...
(3.22)
satisfy the internal transition equations, where
s=
(n,
.. .
, nk-l'
al
ak
and where U satisfies the parametric equations (3.17),
The quantity
(3 .23)
(3.19).
-27-
Q'
C. (sU.)
(3.24)
also satisfies the internal transition equations, whatever the constants
C..
In this section E(s,U) is extended to the boundary-states.
Ex-
pressions are found for all states, but these expressions do not satisfy
all the transition equations.
A small subset of the boundary states
Q remains in.which the error
g(s,U) =
C(s,U)
-
L
T(s,s')(s',U)-
(3.25)
all s'
is not identically zero,
j = 1,...,
(See Section 4.2).
The-e differences g(s, Uj),
' are used to choose the constants C. in (3.24) so that
all the transition equations are satisfied, as shown in Sections 3.4
and 4.2. It is important to note that the set of expressions
derived in this section is not unique.
4(s,U)
It is hoped that this set results
in a relatively compact solution, i.e., that 2 is a small set of
states.
Because of the complexity of the problem and the difficulties described in Chapter 5, attention is restricted to the three-stage line
(k=3).
Even for a system of this size, the complexity makes the manual
generation of the boundary state transition equations impractical.
A
program was therefore written in the IBM FORMAC language (Tobey 1[1969],
Trufyn [n.d.]) which generates these equations symbolically.
The
program listing appears in Schick and Gershwin [1978] and the output
for a three stage line is presented in Appendix A.
Note that transient
states (Section 3.3.1) have been omitted in this listing.
Boundary state transition -equations are defined as state transition
equations in which at least one state (whether initial or final) is
a boundary state (Section 2.4).
There are two kinds of boundary
states in a three-stage line; edge states and corner states.
Edge
states are those in which one storage level is internal and the other
is not, i.e., either
-28-
nl = 0, 1, N1 -1, or N 1
(3.26)
2 < n 2 < N 2 -2
or
2 < n
< N1-2
1-
-5~
1
~~~(3.27)
n 2 = 0, 1, N 2 -1, or N 2
Corner states are those in which neither storage is internal.
n
= 0, 1, N'1
, 'oor N 1
1- 1
1
l(3.28)
n2 = 0, 1, N2 - 1 , or N 2
3.3.1
Transient States
That is, their steady-
Certain boundary states are transient.
state probability is zero.
Consequently, the choice
(3.29)
U(s, U) = 0
suggests itself.
The difficulty in implementing a solution is clearly not in
finding expressions for the transient states; it is in finding the
The procedure is briefly described here and the
transient states.
results are presented in Table 3.1.
There are two kinds of transient states.
The first cannot be
reached from any other state (i.e., the probability of transition
from any other state to that state is 0).
which
.i= 0 and ni -1 = 0.
An example is a state in
Table 2.2 indicates that if
and ni l(t) > 0, then ni_ (t+l) > ni l
ni l(t+l) to be 0 is for ni l(t) = 0.
that if nil(t)=0, the only way for
( t).
i.(t+l) = 0
Thus, the only way for
Table 2.1, however, indicates
i.
(t+l) to be 0 is for ai (t)=0.
Therefore, a state in which ai=0 and nil =0 may only be reached from
states in which a =0 and ni 1=0.
1
If, in addition, ai
1=l
and machine
-29-
(0,n 2 , 0,O,c3 )
n 2 =0,...,N2; a3=0,1
(O,n 2 ,1,0,t
) 3
;
(O,n 2 ,1,1,a 3)
;n2=0,
n2=0,...,N2;
a3=0,1
,N2;
3=0,1
(nl,O,al,O,O)
n =,0..,N1 ;l=,
(n 1
1 ,...,N
nl=0,...
,N1 1; az11=0
=o,1
1 0,ct
1 11,O)
(nl',O,1,'.1)
(ni
(nl,
1
;
,,Ll,0) 1,0,1O) ;;
nl=0,2,3,..',N
1
n
1 =0,2,3,...,N
nl-l,
. . ,N 1
1
(Nl-l,n2,0,1,0)
;
n2=0,...,N2-1
(Nl-1,n 2 0,1,1)
;
n 2 =0,..., N 2 -3,N 2 -2,N 2
(nlN2-lalo,l)
;
nl=0,...,
(Nln2,0,0,o
)3
;
n2=0,..., N2 ; C3=0,1
(N
N1;
1a=o0,1
; a0,1
l,n2,
(Nln2,1,1,0);
n2=0,..., N2-1
(Nln2 ,1,1,1)
;
n2=0,..., N 2 -3, N 2 -2, N 2
(nl,N2,al,0,0)
;
nl=O,*.,N1; al=o,1
(nlN2
l,
o,1);
(nlN2,~l,1,1)
TABLE 3.1
nl=O, ,N1;
..
;
al=o,1
nl=O,...,N1; al=0,1
Transient States
-30i-1 is not starved, this state cannot be reached from another state,
since an operating upstream machine would raise the level in storage
Similar results apply to the case in which ai=0 and n.=Ni.
i-i.
The remaining transient states are reachable only from other transient
states.
For example, consider states in which ni_ 2>0, ai
=l,, ni.l=l,
Table 2.2 implies that at the previous time step, storage i-2 had
ai=0.
one more piece and ni_ = 0.
But this implies that at the previous time
step ai= 0 because machine i cannot fail when storage i-l is empty (see
Table 2.1).
Thus, at the previous step, ni
0 and
.ai=
0, which has
been established as transient.
The full list of transient states, which appears in Table 3.1, can
be established by the following rules.
States where n i-l=
and
.= 0, or a.=0 and n. =N. are
i-l
1
1
1
1
This is because a starved or blocked machine cannot fail
(i)
transient.
(Section 2.3).
States where machine i-1 is operating (i.e., not idle) ni
(ii)
1=
and ai.=; or ai= O, n.= N.-l and machine i+l is operating, are transient.
I
1
1
1
This is because machine i could not have failed if nl
=
1-1
0 or n.=N., and
1
1
an operating machine upstream or downstream would have incremented the
storage by plus or minus one, respectively, in the cycle during which machine
i failed.
(iii)
States where machine i-1 is operating and ni l=0; or ni l=Ni l
and machine i is operating are transient.
This is because an operating
machine upstream or downstream would increment the storage by plus or minus
one, respectively, since by assumption (vi) of Section 2.3, machine transitions precede and determine storage transitions.
It is important to note the following exceptions:
(i)
(1,0,1,1,1) can be reached from (0,0,0,1,1)
(ii)
(1,1,1,1,0) can be reached from (0,1,0,1,1).
(iii)
(N1 , N2-1,1,1,1) can be reached from (N1 , N2,1,1,0).
(iv)
(N -1, N2-1,0,1,1) can be reached from (N -l,N2,1,1,0).
These states are thus not transient.
-31Finally, the fact that machine 1 is never staryed and-machine 3
(for k= 3) is never blocked (Section 2.3) must be considered in deriving
the transient states.
3.3.2
Boundary States Reachable from Internal States in One Step
It is easy to derive E expressions for certain states.
For example,
the transition equation for p(1,2,0,1,1) is
p(1,2,0,1,1) = (1-ri)r 2 r3 p(2,2,00,,0)
+ (1-rl)r2(1-p3)p(2,2,0,0,1)
+ (1-r 1) (1-p 2 )r3 p(2,2,0,1,0)
+
(l-r1 ) (1-p 2 ) (1-p3)p(2,2,0,1,1)
+ Plr2 r3 p(2,2,1,0,0)
+ plr2(1-p 3 )P(2,2,1,0,1)
+ p1 (l-p 2 )r3 p(2,2, 1 ,1 O0)
+
pl (l-p2 )l-P
(3.30)
3 )p(2,2,l,l,l)
All the states on the right hand side of (3.30) are internal.
efficients are the transition probabilities from states (n1 , n 22
The co, l', 2'
a3) to (nl-1, n 2 , 0,1,1), where nl and n 2 (i.e., the initial storage
If (3.21) is substituted into (3.30) and the
levels) are internal.
resulting expression is simplified using the -parametric equations, it is
seen that a natural
choice for E(1,2,0,1,l,U) is the internal form
(1,2,0,1,1,U) = X1 X2 Y2 Y3
(3.31)
Similarly, it can be shown that
f(1, n2, 0,1,1,U) = X1X22Y2Y3
for all 2 < n2 < N2-2.
(3.32)
The relation between (3.31) and (3.32) is generalized
in Section 4.2.
The same reasoning shows that the internal form is an appropriate
choice for the boundary states in Table 3.2.
-32-
(1,n 2 ,0,1,0)
n2
3,..., N2-1
(1,n2,0,1,1)
n
2,...,
=
N2-2
2
(n1l,l,O,,1)
nl = 2,..., N1 -2
(nl,N2 -1,0,1,O)
n1 = 2,..., N1-3
(nl,N 2 -1,1,1,O)
n1 = 2,..., N1-2
1(nl,1,0,0,1)
(nl 1, 1, 0,1)
n1 =
;nl
=
2,..., N1 -2
3,..., N-1
(N1 -l,n 2 ,1,0,0)
n2 =
2,..., N2 -2
(Nl-l,n2,1,0,1)
n2
1,..., N 2 -3
TABLE 3.2
Boundary States Reachable from
Internal States in One Step.
-333.3.3
Other Expressions of Internal Form
The expressions derived in Sections 3.3.1 and 3.3.2 are the only ones
that are determined unambiguously, once the internal expressions are chosen.
The remaining expressions must be derived using guesswork and imagination.
In a sense, there are no correct expressions; the objective here is to
find expressions that satisfy as large a set of transition equations as
possible.
To guide guesses, numerical results were obtained for a case in
which N1 =N2 =10 (Gershwin and Schick [1977]) by iterative matrix multiplications (the power method; see Schick and Gershwin [1978]). It was
observed that several boundary state probabilities, in addition to those
described in Section 3.3.2, appeared to be comparable to adjacent internal
state probabilities.
Generalizing from the case studied, the internal
form is assumed for the states listed in Table 3.3.
3.3.4
Other Expressions
To obtain 5(s,U) expressions for other states s, other approaches
must be used.
One fruitful method has been to look for pairs of equations
that involve the same pair of unknown states, and in which all other states
have expressions already determined.
For example, consider the transition
equations for (1,1,0,0,1) and (1,2,0,0,0). (See Appendix A.)
states are all of the form (1,2,al
1 ,ac2,a
3
).
The initial
Of these, S(1,2,1,0,0,U) = 0
and E(1,2,1,0,1,U) = 0 (from Table 3.1), E(1,2,0,1,1,U) has internal
form (from Table 3.2), and E(1,1,0,0,1,U), E(1,2,0,0,0,U), E(1,2,0,0,1,U),
and -(1,2,0,1,0,U)
have internal form (from Table 3.3).
From these equations, and by performing considerable simplifications
using the parametric equations (3.17) and (3.19), the following
are
obtained:
2
X X2Y
E(1,2,1,1,0,U)
1 2 1
P2
+ p Y)
22
(3.33)
+ p Y )Y
(3.34)
(l-r
2
2
X X2Y
(1,2, 1, 1, 1,U) =
1
2
P2
1
(1-r
-34-
(ln2
°)
;
2=1,.. .,N 2-1
(1,n 2 ,0,0,1)
n2=1,. .,N2-2
(n ,1,0,0,0)
;
(nl1,1,O;
nl=2, . . . ,N1-1
(n1 ,N2 -1,0,0,0)
0
(nl,N 2 -1,1,0,0)
;
(n
N -1,0,O,
21,
(N -l,n2
1
2
(N1 - l,n
2
)
nll....
1
,Nl
nl=2, . . . ,N1 -1
0,0,0)
n 2 1,.
0,0,1)
n2=1, . . .,N
22=
,N2
l ,N2 -1
2
-2
(1,2,0,1,0)
(1,1,1,1,0)
(1,1, 0,0,1)
(N -1,1,0,0,1)
(2,1,1,0,1)
(N 1 (N1
2
,N2 -1,0,1,0)
,N2 -1,1,0,0)
(N-1,N2 -2,1,0,1)
TABLE 3.3
Additional States of Internal Form
-35-
Comparison of the transition equations for (1,1,0,0,1) and (1,2,0,0,0)
with similar equations involving higher storage levels suggests the
generalization of equations (3.33) and (3.34):
XX
n2
2
(l,n2,1,1,0,U)
Y
n2 =2,...,N 2 -1
n
E(ln
2
l'l'l'U) =
P2
(3.35)
(1-r2 + P2Y2)
(l-r2 + P2 Y 2 )Y 3
(3.36)
It must be emphasized that this procedure is more nearly art than
science, and is a way of organizing guesswork which must be verified
later.
According to the procedure and data described in section 3.3.3,
the expression for E(l,n2,1,1,0,U) in (3.35) might also apply for n2 = N 2.
However, there is no justification for this in the transition equations,
and to choose this expression for this state would create additional
non-zero errors g(s,U).
The objective is to maximize the number of
transition equations that are satisfied identically, i.e., to minimize
the number of states for which g(s,U) is not
the parameteric equations (3.17) - (3.19).
zero for all U satisfying
Consequently, E(1,N2,1,1,0,U)
is chosen differently.
Note that E(0,2,0,1,0,U) and ~(0,2,0,1,1,U) expressions were
found using the (1,2,1,1,1) equation, which actually involves the
(0,3,0,1,0) and (0,3,0,1,1) states.
It was assumed, as discussed in
Section 4.2, that
E(0,3,0,1,0,U) = X2 E(0,2,0,l,0,U)
(3.37)
E(0,3,0,1,1,U) = X2E(0,2,0,1,1,U)
(3.38)
Other states whose expressions are found by solving two equations
in two unknowns are displayed in Table 3.4.
The second column of this
-36-
TABLE 3.4
States
Expressions Obtained in Pairs
Generalizations
Equations
n2=2,...,N2-1
(1,2,1,1,0)
(1,1,0,0,1)
(1,n2,1,i,0),
(1,2,1,1,1)
(1,2,0,0,0)
(1,n2,1,1,1), n2=2,...,N2-1
(1,1,0
(1,1,0,1,1,0,0,0)
(1,1,1,1,1)
(2,1,1,0,0)
(2,1,0,1,1)
(1,2,0,1,0)
(n1,,0,1,1),
nl=2,
(2,1,1,1,1)
(2,1,0,0,0)
(n1 ,l,1,l,1),
nl=2,...,N 1 -2
(2,0,0,0,1)
(1,1,01,1)
(n l ,0,0,0,1), nl=2, ...,N-1
(2,0,1,0,1)
(2,1,1,1,1)
(n1,0,1,0,1),
nl=2,...,N -1
(0,2,0,1,0)
(1,2,1,1,0)
(0,n ,0,1,0),
2
-i
n =2,...,N
2
2
(0,2,0,1,1)
(1,2,1,1)
((0,n
2
(0,1,0,1,0)
(1,1,1,1,0)
(0,1,0,1,1)
(0,1,0,1,0)
(N1 -l,2,1,1,0)
(N1 -2,2,0,1,1)
(N -l,n 2 , 1,1,0),
n2=2,...,N 2 -1
(N -1,2,1,1,1)
(N -1,2,0,0,0)
(N -l,n 2 ,1,1,1),
n2=2,...,N -1
(Ni,2,1,0,0)
(N -1,2,1,1,1)
(N
(N1,2,1,0,1)
(Nl,3,1,1,0)
(Nl,n2,1,0,1),
(2,N2-1,0,1,1)
(2,N2-1,0,0,0)
(nl,N2-1,0,1,1),
nl=2,...,Nl-1
(2,N 2 -1,1,1,1
(2,N 2-2,0,0,1)
(nl,N 2 -1,1,1,1),
nl=2,...,Nl-1
..,N1-2
,0,1,1), n 2 =2,...,N 2 -1
2 1,0,), n2-2
'N
-2
n=2,..,N2-2
-37-
TABLE 3.4
States
(Cont'd)
Equations
Generalizations
(1,N2,0,1,0)
(1,N2-1,0,1,1)
(nl,N2,0,1,0), nl
(1,N2,1,1,0)
(2,N2-1,1,1,1)
(ni,N2,1,1,0), n =l,...,N -2
(N-1,N2'0,1,0)
(N -1,N2-1,0,1,1)
(N1 -1,N 2 '1,1,0)
(N -1,N2,0,1,0)
,...,N 1-2
-38-
table lists the equations used, and the last column generalizes these
expressions in the manner described above.
Table 3.5 lists states whose expression can be obtained singly
after express-ions for certain other states are available.
The complete
list of expressions appears in Appendix B.
3.4
Reduction of the System of Equations
A small number of boundary transition equations were used to generate
each expression discussed in Section 3.3.
It is possible to show, using
equations, that these expressions satisfy many of the
the parametric
remaining transition equations identically.
A significant number of boundary equations, however, are not satisfied
identically.
The states that these transition equations lead to are
called "odd" states and are discussed in greater detail in Section 4.2.
The set of odd states is called Q.
much smaller than M.
The number of states 2 in Q is
In this section it is shown how the Markov chain
may be solved by solving a set of Z linear equations, rather than M.
These linear equations are constructed in finding a linear combination
of
(U.) which satisfies all the transition equations, including those
leading to states in Q.
For the Markov chain described in section 2.4,
p = Tp
(3.39)
(T-I)p
(3.40)
or
where I denotes the identity matrix.
For a general k- stage transfer line,
the boundary state probabilities are expressed as a sum of terms, analogous
to the sum of products for internal state probabilities in equation
(3.6):
p(s) = ECj
j=l
i(s,Xlj...
,'
Xk-l,j'Ylj
Ykj
(3.41)
-39-
TABLE 3.5
Expressions Obtained Singly
State
Equation
(0,0,0,1,1)
(0,0,0,1,1)
(1,0,0,0,1)
(1,0,1),0,0,1)
(1,0,1,1,1)
(1,0,1,1,1)
(N1 -1,1,1,1,1)
(N 1-2,2,0,1,0)
(Ni, 0,1,0,1)
(N -1,1,1,1,1)
(O,N 2 0,11,0)
(0,N 2 01,0)
(1,N 2 -1,0,1,1)
(1,N 2 -2,0,0,1)
(N1 -1,N 2 -1,0,1,1)
(N -1,N2-2,0,0,1)
(N1 ,N2 1,1, 0)
(
(Nl,
N 2 - 1,1,0,0)
(N1 ,N2 -11,1,1)
(Nl,N2-1,1,0,0)
(Nl,N2-11,1,10)
1
,N2 ,1,1,0)
-40-
The S(s,U) expressions are derived in Section 3.3.
reduces to
(3.6) when E(s,U)
is given by (3.22).
Note that (3.41)
Following
(3.41),
the probability vector p may be rewritten as
Z'
p
=
E
Cj
_ (Uj)
(3.42)
j=l
where i(U) is a vector whose components are E(s,U).
The number of states
M is given by equation (2.4), and Uj, j =1,..., I' are V' distinct
solutions of the parametric equations.
(T- I)
E Cj(Uj)
Substituting (3.42) into (3.40),
.
(3.43)
j=l
Defining the vector C as
C1
C
(3.44)
C2
and the matrix ~ as
= [(U
1
)
(U2 ) ,,
g(U,)3
(3.45)
equation (3.43) is rewritten as
(T-I);C = 0
(3.46)
System (3.46) has a nonzero solution if the matrix (T-I)E has
rank less than or equal to V'-l.
Equation
(2.14) provides an additional
condition on C through (3.41) which requires that C be nonzero.
A
unique solution is determined if the rank of (T-I)E is exactly equal to
V-l.
-41Because the expressions r(s,U)
satisfy most transition equations,
most components of the vector (T-I)_(Uj) are identically equal to zero
for any U. that satisfies the parametric equations.
For example, in the
three-stage line with storage capacities N1= N2 = 10, 898 components out
of the 968-vector are identically zero.
to odd state entries in i (U).
Those that are nonzero correspond
If 2' is taken to be the number of rows
not automatically satisifed, i.e., Z' =,
then the system of equations
has a unique solution C, once a set of £ distinct U. is chosen.
system in (3.46) can be reduced by computing only those Z rows of
that are not satisfied identically.
(3.46)
The
(T-I)E
The new reduced-order system can be
written-
(3.47)
BC = 0
where B consists of thenon-zero rows of (T-I)S and is.xR
MxM.
This system is analyzed in Section 4.2.
rather than
4.
CONSTRUCTION OF THE PROBABILITY VECTOR
The analytically derived E(U)
expressions
of Chapter 3 are used in
the sum-of-terms solution for steady-state probabilities given by
equation (3.42) to obtain the probability vector.
The system of four parametric equations in five unknowns is discussed in Section 4.1.
The properties of the set of solutions of these
equations are analyzed and an efficient algorithm for solving these
The reduced-order system of equations (3.47)
equations is presented.
derived in Section 3.4 is studied in Section 4.2.
B matrix is analyzed.
The structure of the
A singular value decomposition solution is dis-
cussed.
The limiting behavior of the solution sets of the parametric equations
After these limits are found, they are
is described in Section 4.3.
used in deriving limiting
(UW) in Section 4.4.
constructing a better behaved B matrix.
These are utilized in
The numerical implementation
of these results is discussed in Chapter 5.
4.1
Analysis of the Parametric Equations
By analyzing internal state transition equations and guessing a
sum-of-products solution for internal state probabilities, the following
parametric equations are derived in Section 3.2:
k
H
(-r i + piYi)
(4.1)
1
i=l
X.Y.
X---_Xj- 1
Here, X
0
r. + (l-p.)Y.
J
+
Y_
-r
+
Yj
j = 1,...,k
= Xk = 1 are defined for notational convenience.
stage transfer line (k=3), equations
(4.1) and
(4.2)
For a three-
(4.2) comprise a set of
four nonlinear equations in five unknowns.
As discussed in Section 3.4, a subset of R'distinct solutions of
these equations may be used to compute the probability vector.
-42-
However,
-43it is shown in Chapter 5 that because of numerical problems, certain
sets of solutions are better than others.
For this reason, the full set
of solutions is analyzed in this section, and some important properties
are discussed.
Internal state probabilities have the form
nl
p(s)
1
j.
X
k-1
..=C.X
Xk~i'.
1
a
1 ...Ykj
(4.3)
j=l
The exponents n. take integer values such that
2 < ni < N
-2
i = 1,...,
k-l
i.e., n. takes on odd as well as even values.
(4.4)
For p(s) not to oscillate
with period 2 with ni, it is necessary that
X..
> 0
(4.5)
all i,j
The fact that the probabilities do not oscillate agrees with intuition, observation of computer simulation (Schick and Gershwin J19781)),
and a numerical solution of the transition equations (Gershwin and
Schick [1977]).
In order to make the study of the solution sets clearer, a relationship is sought between Xlj and X2j,
subscript notation (X1 , .. .
Xk_,1
Reverting for simplicity to the singleY1'
' Yk) of Section 3.2, the variable
Q.j is defined as
X.
j = 1,..., k
Qj =
(4.6)
j-1
Equation (4.2) is solved for Yj in terms of Qj.
Using the quadratic
formula, it follows that
Y. =
[1-p.-(l-r.)Qj] + /[1-pj-(l-rj)Q.]
+ 4rjpjQ.
J
J-...
....
a2j
2p Qj
(4.7)
-44Equation (4.7) can be substituted into (4.1), giving a single equation in
the two unknowns, X1 and X 2 , for a three-stage line.
not explicitely stated here.
This equation is
Generally, in a three-stage line (k=3),
for any set of failure and repair probabilities Pi and ri such that
1-pi-r. > 0, i = 1,2,3, the full set of solutions of equations (4.1)
and (4.2) with non-negative X. is given by a set of seven curves similar
to those illustrated in Figure 4.1.
(The conditions l-pi-r i > 0 is not
restrictive, since for most realistic systems the probability of a failure
or a repair during a single machine cycle is small.)
That these seven curves comprise the full solution is demonstrated
by using the following lemmas and Lemma 4.5 of Section 4.3.
Lemma 4.1:
The minus sign is equation (4.7) yields negative Y..
The
plus sign in equation (4.7) yields positive Y..
Proof:
Since Xj are non-negative, so are Qj.
are probabilities and hence non-negative.
[l-pj-(l-rj)Qj] <
Furthermore, pj and r.
It follows that
Il-p-(1-rj)j]
+ 4rjpjQ
(4.8)
Since the dencominator in (4.7) is non-negative, equation (4.8) proves
the lemma.
Different combinations of signs in equation (4.7) give the different
curves in Figure 4.1.
Since k=3, there are 8 possible sign combinations.
However, not all combinations satisfy equations (4.1) and (4.2) together
with the requirement that Xj be non-negative, as shown in Lemma 4.2.
Lemma 4.2:
Of all possible sign combinations, only those which include
at most one minus sign satisfy equations (4.1) and (4.2) and the requirement that Xj be non-negative.
That is, either Y 1 , Y2 Y3 are all
positive, or else exactly one of them is negative.
Proof:
Substituting (4.6) into (4.2),
-45-
t
3.0
6
-l66
4
4
-7
2.6
2.4
2.2
t2
0.2
1-
0
00
0.2
62 f
0.4
0.6
~.6~5~X
0.8
1.0
1.2
1.4
1.6
.8
2.0
2.2
2.4
2.6
2.8
3.0
xI
Figure 4,1;
Locus of
(Xl,X 2) parameters,
curves, a set of (Y1,Y,Y
2
For every point on these
3 ) exists
such that (XlX ,Y
2 ,1
Y2,Y3) satisfies equations (4.1) and (4.2) for k=3.
For this plot, the failure and repair probabilities are:
P1 =.l, p2 =p3=. 05, rl=r 2 = .2, r3=.15.
The large numbers
indicate the curve, which should be compared with Table
4.3.
The small numbers indicate the limit case, which
should be compared with Table 4.5
-46r. + (1-p.)Y.
=
j
J.
(4.9)
(49)
Yj(1-r. + pjY)
The parameter Z. is defined as
J
Zj = 1-rj
Using (4.10),
=
+ pY.
(4.9) becomes
(1p
~-J-
Qj =
(4.10)
z. -'a
)
1,pj
(4.11)
_______________
Z.
Z.
(l-r.)
Equation (4.1) may be rewritten as
3
H[
(4.12)
Z. = 1
i=l
while it follows from (4.6) (where X 0 = X 3
3
=
1) that
(4.13)
i - 1.
i=l
Defining the variable %
as
(4.14)
Wj = QjZj
it follows from (4.11) that
--=(42)
(4.1)
Equations
1-j
i-(4.15)
(4.12) - (4.14) imply
= 12,3p
j = 1,2,3
-47-
3
7
Wi = 1
(4.16)
.
i=1
Thus, there are now five equations,
unknowns, Zi and Wi, i= 1,2,3.
(4.12),
(4.15) and (4.16), in six
These are equivalent to the four equations,
(4.1) and (4.2), in five unknowns, X., i = 1,2 and Y., j - 1,2,3, through
)
1
transformations (4.6), (4.10) and (4.14).
The graph of equation
(4.15) is plotted in Figure 4.2.
three such graphs, for j = 1,2,3.
There are
The W.- and Z -intercepts and asymptotes
appear in Table 4.1.
Equation (4.15) can also be written as
j1-r -pj
1-r.
Zj = (1-rj)
Wj - (1-pj)
%
(1))]
(4.17)
Since Qj > 0, equation (4.14) implies that Zj and W. have the same
sign.
However, if Z. < 0, (4.15) shows that W. > 0.
W. < 0, (4.17) indicates that Z. > 0.
Similarly, if
Therefore, Z. and W. must both
be positive.
The sets A, B, and C are sets of (Zj, Wj) pairs that satisfy equation
(4.15) and fall in certain intervals.
Table 4.2.
These intervals are indicated in
Note that
1-r. -p.
1- r<
1-pj
1--r.j < 1
3
and
l-r. -p.
<
r
1- r.
for rj > 0, pj > O.
Y. > 0
Y. < 0
3
+
l-p. < 1
It follows from equation (4.10) that
Z. > 1 - r.
3
++ Z. < 1 - r.
-48-
1.5
I
I
I
I-rI
I
iip;
Y0.5O - A
Zj
Figure 4.2:
Graph of equation
(4.15)
jPjI
-Y. >O
for pj=.l,
I
r.=.2
j
-49-
Z.j
Wij =
l-r . -p.
-r.
Zj + X
Wj+
l-pj
Z.
=
J
W. = 0
1-pj
Z. +1-r.
TABLE 4.1:
J
W. +0
Limiting Values of Z. and W.
)
J
-50-
Interval
Set
0 <Z'
Z
A
< l-rj-pj
j--
O< W<.
j
-
1-
1-pj
-r
l-r.
-
r. < Z. < 1
B
1< w. <
1
<
Z.
<
b
1-pj < W. < 1
TABLE 4.2:
Bounds on Sets A,B, and C,
-51Note that W. =Z.j 1, j= 1,2,3 satisfies (4.12), (4.15), and (4.16) and
J
J
Y > 0, j= 1,2,3.
Except for that point, equations (4.12) and (4.16)
imply that
(i)
(ii)
Z. > 1 and
1
Z. < 1 for at least one value of i and one of
J
Wi, > 1 and W., < 1 for at least one value of i' and one of
-J
i',
i'/
j'.
Statement
(i) implies that (Zi , W )
i s C for at least one value of i.
Statement
(ii) implies that
(Zi, Wi.)
s B for at least one value of i'.
Therefore, there must be at least two pairs in BUC and at most one pair
in A.
Since Y. > 0 in BUC and Y. < 0 in A, the lemma is proved.
The curves in Figure 4.1 correspond to the sign combinations in Table
4.3.
The identification of sign combinations with curves is established
by examining limits and intercepts.
(See Section 4.3.)
By means of the Wi, Z i equations, it is possible to efficiently generate
U points.
Equation (4.15) may be substituted into
(4.16), giving an equation
in Z i , i = 1,2,3:
l-r-p
W=
(l-P
(4.18)
so that
J(1l-1-P=
(1-pj)
j=l
Using equation
1=1
Zj - (1-_rj)
(4.12),
= -p)(-p)-(1-)
(4.19)
= 1(
(Z Z 3)
13s
3uto
is substituted for Z2 in (4.19), giving
2
. Zzl
r
(l-r
I3(l)
(r
2
)
fZ 3
]
-
(1-r
3
)j
4
]
(4.20)
-52-
Signs
Curve #
.
1
I
_Y
-
2
+
3
+
2
Y
~+
+
+
+
4
+
6
+
+
+
+
TABLE 4.3
+
+
5
7
Y3
+
Sign Combinations for the Curves on Figure 4.1.
-53-
If either Z1 or Z3 is assigned a value, equation (4.20) reduces to a quadratic equation in the remaining variable.
For instance, (4.20) can be
written
2
2
2
+ ]BZ
Z 3[AZ
+ Z+3[CZ
+ DZ 1 + E] + [FZ
+ G] = 0
(4.21)
where
A =
2 -
2
.
B = 01~2
-
12
2B3 - 72Y3
C =
D = a(l-S 1 283) - (1 -
E = y
(4.22)
1Y2Y3 )
2
- a
F = Y3 G = ~1
3
3
-
lY3
and
3
a=n(l-Pjp
j=l
1-r -p.i
i
=
i
=
1,2,3
l-Pi
= 1- ri
Yi
1
1
(4.23)
i = 1,2,3
In this manner, solving a system of four non-linear equations is reduced
to solving a quadratic equation, since Y. is uniquely determined by
Zi and Xi by Yi, using equations (4.10) and (4.2), respectively.
1
1
~~1
-54-
The use of each solution Uj = (Xlj, X2j
Ylj' Y2j' Y3j ) in computing
the probability vector is discussed in Chapter 5.
To put this implementa-
tion in context, however, it is necessary to analyze the reduced-order
system of equations (3.47).
4.2
This is done in Section 4.2.
Analysis of the Reduced-Order System of Equations
The reduced-order system of equations (3.47) is obtained in Section
3.4 by deleting all the zero rows from the matrix (T-I)E.
As stated in
Section 3.3, only a small number of E(s,U) expressions do not satisfy
Thus, most rows in (T-I)E vanish, and the
all transition equations.
order of the system of equations is drastically reduced.
Set 2 is the set of states s such that
g(s,U) = S(s,U) -
T(s,s')E(s',U)
(4.24)
all s'
is not identically zero.
That is, Q is the set of states corresponding
to those entries in the vector
g(U) =-
(T-I)_(U)
(4.25)
which are not zero for all solutions U of the parametric equations (4.1)
and (4.2).
Since these equations are derived from internal state
transition equations, it follows that g(s,U) = 0 for all internal
states.
Furthermore, the expressions C(s,U) are chosen so that g(s,U) = 0
for almost all boundary states s.
ary states.
Thus, Q is a small subset of the bound-
Transition equations corresponding to odd states (i.e.,
those in which the final state is an odd state) supply the information
necessary to compute the weighting and normalizing constants Cj of
equation (3.41).
Set Q is displayed in Table 4.4.
Odd states may be subdivided, as mentioned in Section 3.3, into two
groups: edge states, and corner states.
The number of odd edge states
depends on the storage sizes; the number of odd corner states is constant.
The total number Z of odd states, which may be obtained from Table 4.4
by inspection, is
-55-
(N1 -1,N 2 -1,1,1,1)
Corner
States
(1,N 2 -1,1,1,1l)
(N1 -1,N 2 ,1,1,0)
(0,1,0,1,1)
(N1,0,1,0,1)
(nl,0,1,0,1)
;
(nlOlOl)
-2 n
<1-N1
2 < n 2 < N2-1
(0,n2,0,1,0)
Edge
States
2 < n1 < N -1
(O,n2,0,1,1)
2
;
2 < n- 2 -< N- -
(N1ln21,0,0)
;
1 < n 2 < N2-2
(Nln ,1,0,1)
2
1 < n2 < N2-2
(nl,N 2 ,0,1,O)
1 < n
< N1-2
1 < n
< N1 -2
(nl,N
TABLE 4.4:
2
,1,1,0)
;
Q, The Set of Odd States
-56-
(4.26)
= 4(N+N)- 10
As stated in Section 3.4, I is linear in storage sizes, while the total
number of system states (equation 2.4) is quadratic in storage sizes.
Thus, the reduced-order system of equations (3.47) increases in dimension
more slowly than the original system, (2.19).
The matrix B has a special structure.
Efficient algorithms to solve
This has, however,
(3.47) may be found if this structure is exploited.
not yet been done.
After possibly reordering the odd states s (corresponding to rows of
B), the B matrix has the following form:
B0
VlG
VlG12
VlG13
VG14
B=
(4.27)
VG
2 21
V2G22
V2G23
V2G24
The block B 0 is a matrix corresponding to the corner odd states.
Its dimension is dxl', where d is 14.
It is not 6 (the number of odd
corner states) because some of the odd edge states have error expressions
which do not fit into the scheme described below.
The number d is con-
stant and does not depend on the storage sizes N1 and N 2.
To describe the other blocks in B, it is observed that edge state
expressions obey relationships of the fQor
S(n
1
+ n,
n 23'
2
,
2
'
Uj) =
Xlj(nl, n2, al'
2
,3'
'
Uj)
(4.28)
-57where n2 obeys (2.23) or (2.24) and nl and nl+n obey (2.25).
1(nl'
n2+n' a 1' 2'a3'
2j(n,
n2, al
Uj) = X
' a3
)
where n1 obeys (2.23) or (2.24) and n 2 and n2 +n obey (2.25).
Also
(4.29)
It must
be noted, however, that in some cases, after equations (4.28) and (4.29)
are assumed, it is found that similar relations hold for storage levels
equal to 1 or Ni-l also.
Thus, for instance, the expressions ;(nl,O,O,O,l,U)
may be written
(2,0,0,0,1,U.)
U(3,0,0,0,l,U)
2.
= Xlj
(2,0,0,0,l,U.)
l
N1-4
E(N1-2,0,0,0,1,Uj
) = Xlj
(4.30)
U(2,0,0,0,l,Uj
N 1 -3
where E(2,0,0,0,1,U) is given in Appendix B.
Each element in B is given by g(s,U) as defined in equation (4.24).
Each g(s,U) which corresponds to an odd edge state may be shown to obey
relation of the form of equations (4.28)- and (4.29).
Thus, for example,
n -2 r
g(nl,O,O,O,l,U) = Xl
IE(2,0,0,0,1,U)
-(l-r 1 ) (l-r2)r3 (2,1,0,0,0,U) - (l-r) (l-r2 ) (1-p3)r(2,1,0,0,1,U)
-(1-rl)p2 (1-p3 )P(2,1,0,1,1,U) - pl(l-r2 )r3
-Pl(l-r2)(-P3)(2,1,1,0,1,U)
-(l-r 1 ) (l-r 2 )E(2,0,0,0,l,U)
- plP2 (1-P
) (2,1,1,1,1,U)
- pl(l-r 2 )(2,0,1,0,1,U)],
nl = 2,..., N1-2
This relationship is only valid for n
n
= N -l because U(nl(2,l,l,l,l,
)
n -2
X1
(2,l,l,',0,U)
(4.31)
= 2,..., N1-2 and not for
1
(2,1,,1,1,U) for nl= 2,...,N12
-58-
only,
This is why g(N -l,l,l,l,l,U)
is part of B0 and why the dimension
of B 0 is dXZ' where d is greater than 6 (i,e., d= 14).
This leads to a relation similar to equation (4.28),
n -2
g(nl,O,O,O,l,U) = X 1
g(2,0,0,0,1,U), n1 = 2,..., N 1 -2
(4.32)
Equation (4.32) suggests that the elements in B corresponding to these
odd edge states may be rewritten as a product of two matrices, V1Gll,
where
V
=
1
1
Xll
X12
Xl
2
Xll
2
X12
2
Xl'
N1-4
. . .
N-4
X11
X12
1
1
(4.33)
-4
. .
X
a Vandermonde matrix (Bellman [1970]) of dimensions (N1
g(2,0,0,0,1,U1 )
0
G
11
=
-
0
g (2,0,0,o,Q
0
0
0
0
3)X'
and
0
,
2)
.
0
is diagonal, of dimension 'x
.
g(2,0,0,0,1,U,,)
(4.34)
The elements corresponding to the states in the second, seventh,
and eighth rows of the odd edge states in Table 4.4 may be rewritten as
similar matrix products, with the same V 1 post-multiplied by different
Gli, i = 2,3,4.
Similarly, the third through sixth rows are rewritten
-59-
as products V2G2i, i = 1,..., 4 where
1
X21
v 2 =.
. . .
1
1
X29,
X22
.
222
N 2-4
N 2-4
X22
X21
(4.35)
N2-4
.
.
X2z
and the matrices G2i are appropriately defined diagonal matrices.
From the reduced-order system of equations (3.47) it is clear that
the B matrix must be singular, since C f 0 if equation (2.14) is to be
In Lemma 4.3, it
satisfied.
'isshown that, for
It is believed that for '" > I, if the U. are all
is no greater than X.
distinct, the rank of S is exactly A.
B of
shows that the rank
If A' >
Lemma 4.3.
Proof:
Let (T-I)
-,
the rank of i
l'>
is R-1.
Under this assumption, .Lea 4.4
In the.sequel, Qu
Q.
,..
the rank of S is at most Z.
be the jth row of T-I.
Matrix E has been constructed
so that exactly M-k of the M rows of T-I satisfy
(T-I).
=
0 .T
=
(4.36)
Form the matrix T' by deletina one of the other rows of T-I and replacing
T
it by V = (1,1,...,1). Again exactly M-t rows T! of T' satisfy
T'=
.
(4.37)
T
T.
(4.38)
0
This is because
V
T,
which is possible to verify, tediously, using the internal expressions
(3.22), the boundary expressions in Appendix B, and the parametric
-60equations (4.1),
(4.2).
(It is easier to do what the authors did: to
verify this numerically.)
Matrix T' has been constructed to have rank M.
row vectors in equation
Therefore the M-k
(4.37) are linearly independent.
are orthogonal to these vectors.
The columns of E
They therefore lie in a subspace of dimen-
sion < Q, and the lemma is proved.
Lemma 4.4.
Proof:
Assume
. '= Z and the rank of 2 is Q.
The rank of T'I
is L (Hadley [1964],
Then the rank of B is i-1.
page 139).
Therefore, the
rank of (T-I)E is k or Z-l, since it is the same as T'E except for one
row.
If e is a nonzero vector of dimension Q, there is a unique solution
C to
(4.39)
T'EC = e
which isnot zero.
T
Let e be
(0,...,0,1,0,...,0), where the non-zero
element is in the location corresponding to the V
row in T'.
Then
(T-I)EC = 0
(4.40)
since the row that is replaced in T-I by V
is linearly dependent on the
other rows (because T-I has rank M-l).
Therefore, the rank of
non-zero rows of
(T-I)5 is Z-1.
Since B is composed of the
(T-I)-, B also has rank Z-1, and the lemma is proved.
This property is used in Section 5.1, where the numerical problems
caused by solving
(4.40) by singular value decomposition are discussed,
and a procedure to avoid such problems for at least moderately large
storage capacities is introduced.
4.3
Limiting Behavior of U
It is shown in Section 4.2 that the reduced-order system matrix B
may be partitioned into submatrices.
Some of these are products in-
volving Vandermonde matrices, which are known to be poorly behaved
(Bellman [1970]), and may be partly responsible for the difficulty in
-
----
- ·- ·- ·· ··- ·----
·- ·--
·---
·-
- · ·---
·-----
--- · ·- · ·· ·· ;· ;--r
· ·- 11 ··
I'---·--·- ·· ·- ·
------
· ·----
·- ··
-61treating. B (see Section 5.1.)
Furtherxore, this difficulty is exacerbated
as N 1 or N2 (and hence the dimensions of B) increase.
The behavior of the vector i(U) as U approaches limiting values is
analyzed with the motivation of rendering the B matrix better behaved and
easier to construct.
The limiting values of U are obtained by allowing
Qj (i.e., X1 , X2/X 1 , or 1/X2 , for j = 1,2, and 3 respectively in equation
(4.6)) to go to zero or to infinity for some values of j.
Once these are
analytically derived, the limiting i(U) are obtained by substituting the
limiting U into E(°) and scaling so that the vector is non-zero and bounded.
Limiting V(U) vectors
The derivation of limiting U is described below.
are derived in Section 4.4 and a complete listing appears in Appendix C.
As a first step, the limiting Z. corresponding to limiting Qj are
derived.
Equation (4.11) may be rewritten as a quadratic equation in Zj,
- Z [Qj(l-rj)
Z2jQ
(4.41)
1-p] + 1-rj-pj = 0
which, using the quadratic formula, yields
EQ.(1-r.) + 1-p.
+
+ 1-p.] 2 - 4Q.(l-r.-p.)
+ /[Q.(l-r
j
2Qj
(4.42)
+
+
+
If Yj is obtained from Zj by equation (4.10), then Yj > 0 and Yj < 0.
The case where Qj
+
X
The root in equation
is analyzed first.
(4.42) may be rewritten as
=
([Qj(l-r)
- (l-pj)] 2 + 4Qrp)u/2
(2
J=([+
Qj ((1-rj
-
]
(1-pj)Cj
(4.43)
+4r.p.
+
4 4
1/2
p
)
+ 4rip..) 1 / 2
(4.44)
-62-
where by definition,
and Ej
+
(4.45)
Qj
J
0 as Qj
+
i6.
= Q
o. Equation (4.44) may be expressed as
Ef )
-
and using a first order Taylor expansion around
6-
Q [fj (j')
(4.46)
= 0,
0.
Ej.
()+
(4.47)
+ f'(0)
= Qjf(0)
(4.48)
where from equation (4.44),
f(O) = 1-r.
(4.49)
,-r.-p.-r.p.
j
l-r.
f'(0) =
(4.50)
Thus, equation (4.48) becomes
l-r.-p -r.p.
6. Q J(-r.)]
-
1-r.
Equation (4.51) is substituted into (4.42), giving, as Q
+ ~
Z
+
o,
+ 1.-r
.
1-r. +
(4.51)
(4.52)
l-r.-p.
Z3
O
(4.53)
J
Qj.(1-rj)
The case where Qj
+
0 is analyzed similarly.
Expanding the right
hand side of (4.43) in a Taylor expansion around Qj = 0,
that
it follows
-63-
(l-r.-p.-r.p.)
6j
-pj
l-p.
Z.j
(4.54)
(4.42) yields, as Qj
Substituting (4.54) into
Z.-,
.j
-Q
i-p
j
+
0,
r.p.
+
Q.
(4.55)
+
1-p
l-r.-p.
J1-pj
(4.56)
The limiting relations between Zj and Qj for each value of j are
given by equations (4.52),
(4.53),
(4.55), and
(4.56).
The relations
among Qj and Zj, for j= 1,2,3 are now analyzed.
Equations
(4.53) imply that as Q.
(4.52) and
(Zj); equations
finite value (Z.) or zero
as Qj
+
+
, Zj approaches a
+
(4.55) and (4.56) imply that
0, Zj approaches infinity (Zj) or a finite value (Zj).
3
3
Since
+
Y. is obtained from Z. by equation (4.10), and Y. < 0, Lemma 4.2 inpoges
restrictions on which combinations of limits satisfy the parametric
equations.
Specifically,
Zj + Zj for at most one value of j.
Investigating the limiting behavior of U as X. + 0 or X. + X for some
1
3
values of i and j is equivalent to investigating the behavior of U as
Qi' i = 1,2,3 approaches limits (equation (4.6)).
It is also equivalent
to deriving limiting U as Wi , i = 1,2,3 approaches limits.
Using this
latter approach, the limiting values of U are completely characterized
in the following lemma:
Lemma 4.5:
There are 12 sets of limiting U =
Zj and W., j = 1,2,3, corresponding
(X1, X2 , Y1 , Y2, Y3 ).
(through equations (4.14),
The
(4.6),
and (4.10)) to these limits are given by:
(a)
For each permutation
w
(a,~,y) of the numbers
(1,2,3),
0
(4.57)
l-r-p
-64w
+
-*.
(4.58)
Z~
1-r
+
(1-l-p)
(p-r-p)(1-r
(1-p)
)
(1-r
-p )
i
.
W +
Yr
t
a(1-1-p-
Pa) (1-r)(1-ry)
(4.59)
1-ip
(1-ra-po) (-r
(b)
For each permutation
z+
)
(a,a,y) of the numbers (1,2,3),
0
(4.60)
l-r-p
a
1-r
zg '
Z+
I
l-pS
(4.61)
(1-r-) - (1-r(1-()r,-p)
( 1 -PO (1)y
-p)(1-p)(1-p)
1-r,
Y (l-zj-)
~
(4.62)
l-ra
W +
Proof:
(a)
If both W
-r
W +
If Wa + 0, then W W
and Wye-,
Y
then equation
for equation
(4.16) to be satisfied.
(4.15) is used to give
-65-ra-pa
+
Z
1-r
Z-+
Z
Y
+
1-r
(4.63)
B
Y
These values cannot satisfy (4.12) since they are each less-than one.
only one of (WB,W ) may approach infinity.
Assume W i.
Then, Z
Thus,
and Z8
are given by (4.15) and appear in (4.57) and (4.58). Finally, Z is chosen
-1
to satisfy (4.12), and is (ZaZ)
. Note that the expression for Wy is
finite and positive.
The denominator is positive since
i-pa
(4.64)
(l-ra-pa)(l-rY-p)
(4.65)
(l-r-p) (l-r ) (l-ra) < l1-r-pc
<
and the numerator is positive because
(l-r,-pa)(l-r )(l-r -p.) <
(4.65)
< (l p(X) (l-PY)
Case (a) covers six cases, for each permutation of (1,2,3).
Case (b)
covers the remaining six cases, and are treated in exactly the same way.
Lemma 4.5 is thus proved.
To expressthese results in terms of X1 , X2, Y1, Y 2, and Y3 , equations
(4.10), (4.6), and (4.14) are used.
Some symmetry is lost,
=
since Q1
X1,
Q2 = X 2 /X1 , and Q3 = 1/X2 . For example, in part (a) of lemma 4.5, (4.14)
implies that Q%+0, Qgi~, and Q approaches a nonzero, finite constant.
If (c,~,y)
= (1,2,3), then XF0, X /X1 + a, and X 2 approaches a nonzero
finite constant.
X2+0,
If, on the other hand,
(a,B,y) = (1,3,2), then X 1
and X2 /Xlapproaches a nonzero, finite constant.
quite different cases, as discussed below.
(b) implies that
Qa+X, Q+O, and 9
+
0,
These are two
It may also be noted that part
approaches a nonzero, finite constant.
Although this appears to be merely a different ordering of (a,c,y), in fact
the cases are different because the constant terms differ.
limit combinations in terms of X i and Qj appear in Table 4.5.
The possible
The last two
-66-
-1
Permutation
((,Y)
Part
Case
Xl=Ql
1
0
Constant
0
(3,1,2)
b
2
0
Constant
0
(1,3,2)
a
3
0
o
Constant
(2,1,3)
b
4
0
o
Constant
(1,2,3)
a
5
Constant
0
0
(3,2,1)
b
6
Constant
0
0
(2,3,1)
a
7
Constant
o
0
(2,3,1)
b
8
Constant
o
~
(3,2,1)
a
O0
Constant
(1,2,3)
b
O
Constant
(2,1,3)
a
Constant
o
(1,3,2)
b
Constant
o
(3,1,2)
a
Case
9
10
11
12
co
TABLE 4.5
x2/x= Q2
Q
X2
Limiting Qj Combinations.(Note that Qj
are related by equation (4.6).)
and X.
The "Constants"
are finite, positive real numbers which are generally all different.
Compare with Figure 4.1.
-67columns of Table 4.5 indicate the part of Lemma 4.5 and the permutation
that each limiting case corresponds to.
Case 1:
Some examples are analyzed below.
X1 and X 2 approach zero, and the ratio X 2 /X1 approaches a constant
(see Figure 4.1, curves 1 and 6).
The limiting Qj are the following:
Q1 -+ O
Q2
(4.66)
constant
+
Q3 + 0
From equation (4.55), it follows that if Q1 + 0, then either
+
Z1
.
Z1
Z
~
1-r
Z1
1
Z~~~~~~~~+
00
~~(4.67)
-P 1
(4.68)
Z1 =
~1
1-p
1
must
If Z -+Zi+, as in (4.67), then at least one of the remaining Z.
I
11
approach zero, in order to satisfy equation (4.12). Since Q2 + constant
implies that Z 2
+
constant (see equation (4.42)), it follows that
(4.69)
Z3 + Z3 = 0
Then, by Lemma 4.2, it follows (using (4.12)) that
Z+
2
z2
(4.70)
1
+-
1 3
Substituting the first-order Taylor expansions in equations (4.53) and
(4.55),
Q3 (1 -r3 )
Q1
z+
=
-
(4.71)
1l-r 3
lr
(4.72)
-68where equation (4.13) has been used.
two unknowns, Z2 and Q2 . Equation
two unknowns.
Equation
(4.72) is an equation in
(4.42) is an independent equation in these
Substituting (4.72) into
(4.42) and simplifying, it follows
that
(I-p
)(l-r3-1 (1-p
)[(3
2
(1-Pl)-r3-P3) (l-r 2 )
3
p2 32
) (1-r
l-r3)
3
(1-Pl)(l-r2 -P2)(l-r3-P3)]
(4.73)
(l-r2)(l-r3)-
2
Z+
2
(1-p
3
l-r 3 - (1-
1)
1
- 2
) (l-r2
2
P
2)
(1-r
3
3
)
(4.74)
3
(1-P2)
(ll-r3_P3)
The limiting U follows from equation (4.66), as well as (4.67),
and (4.10).
As Z
, Y1
+
+
I.
(4.69),
If Z3 + 0, then from (4.10),
P3
Y
3
=
(4.75)
3
l-r
3
Y2 is found by substituting (4.74) into
implies that Xl
+
0, X2
+
(4.10) and solving, while
(4.66)
0, and X2/X 1 approaches a nonzero, finite con-
stant.
Note that these results could also be obtained, as indicated by
Table 4.5, by using equations (4.60) - (4.62) with (cL,O,y) =
(3,1,2).
To obtain the limiting Xi, Yi, and Qi' transformations (4.10),
(4.14),
and (4.6) are applied.
Case 2:
One the other hand, if Z 1
+
Z1 as in (4.68), then since Z1 is
nonzero and finite (as is Z2 ) , it follows that Z 3 must approach a
zero and finite limit.
Z3
3Z3
23
=
From equation
non-
(4.52),
lr 3
3
(4.76)
Furthermore, Lemma 4.2 requires that Z2 + Z2, so that using (4.12),
Z2
2
(Z3
(4.77)
1-p 1
( l-r3 ) (1-r
l
-P)
-69-
which determines Y
.2
Also, substituting (4.68) and (4.76) into (4.10),
it follows that
rY
1
(4.78)
1-p 1
Y3 = 0
(4.79)
The limits of X1 , X2, and X2/X 1 again follow from (4.66)
and are 0,0,
and a nonzero, finite constant.
A complete listing of limiting U appears in Appendix C.
These limits,
or Lemma 4.5 and Figure 4.2, may be used to characterize curves 1-6 in
Figure 4.1.
In part (a) of Lemma 4.3, the pair (Za,Wa) must be in region
A of Figure 4.2, as it approaches limit (4.57).
Thus, Y,<0.
Moving
continuously in region A, (ZcaIWa) approaches (4.60) in case (b).
If
a=l, then X 1 = Q1 = Wl/Z1 (from (4.14)) varies from 0 to infinity.
If R2in
part (a) and y=2 in part (b), then X /X1 = Q2 varies from infinity to a finite
value, so that X 2 varies from a finite value to infinity.
that this characterizes curve 4 in Figure 4.1.
It is clear
The other five unbounded
curves.may be similarly described.
Finally, the fact that curve 7 is bounded and is in the neighborhood
of (X1 , X 2 ) = (1,1) when r.i and Pi are small may be demonstrated as
follows.
If all Y.
> 0, then Z.1 > 1-r.1 for all i.
1
(Zi. W i ) e BU C in Figure 4.2. For any permutation
(1,2,3), Z
> l-r
This implies that
(a,E,y) of the numbers
implies that
(4.80)
Za < (l-r) (-r
from equation (4.12).
W.i > 1-Pi for all i.
Similarly, if (Z., W)
Thus, from (4.16), W
Wc<
Then, (-psince)
(-p
Then, since
e
B U C for all i, then
> 1-pc implies that
(4.81)
-70W
zQa
(4.82)
it follows that
(l-pa) (-r
B ) (l-r)
<
<
(4.83)
-p
(4.83) implies that Q~ is in the neighborhood of 1.
For small Pi and ri,
Since the same development applies to any permutation
(a,O,y) of
(1,2,3),
it follows that X1 , X 2,and X 2 /X 1 are bounded and are in the neighborhood
of 1.
4.4
This characterizes curve 7 in Figure 4.1.
Limiting Behavior of i(U)
The scaled limiting E(s,U)
(henceforth denoted by Ek(S),
where
k = 1,...,12 denotes the limiting cases derived in Section 4.3) is described in this section.
limiting vector E
k ( s) # 0
(U) is scaled to ensure that the
The vector
is nonzero and bounded.
for some s
k
|ik ( s )
I
That is
Ik-1,.,.
12
(4.84)
< - for all s
The conditions expressed by (4.84) are achieved by performing the
following operations:
(i)
If a variable that approaches zero occurs in the numerators
of all elements of
(-) divide by the lowest power that
the variable is raised to.
(ii)
If a variable that approaches infinity occurs in the numerator of at least one element of _(-),
divide by the highest
power that the variable is raised to.
(iii)
If a variable that approaches zero occurs in the denomi-
nator of at least one element of E(-), multiply by- the
highest power that the variable is raised to.
-71(iv)
If a variable that approaches infinity occurs in the
denominators of all elements of E(.), multiply by the
lowest power that the variable is raised to.
It is important to note that in scaling i(-), the direction of
the vector is not changed, and that is all that matters.
The change
in the magnitude of C(.) is compensated for by the magnitudes of the
elements of the C vector (i.e., the weighting and normalizing constants
of Section 3.4).
In performing operations (i) - (iv), it must be borne in mind that
certain products or ratios of variables which approach zero or infinity
For example, in case 1 of Section
may themselves approach constants.
4.3, X 1 + 0 and Y1
+
o.
From equation (4.2)
(with j = 1), it follows
that
r +
-r
X1Y. =
as Y1
+
(l-pl)Y
+ p Yl
1 + P 1Y 1
1
1-p
(4.85)
p1
o, giving a nonzero, finite constant.
Similarly, X2 always
appears in the numerator of expressions in which Z3 appears in the
(See Appendix B.)
denominator.
Thus, from equation
(4.2)
(with j~3),
it follows that
Y
+ (l-p)3)Y3
r +
rY
(1-P)Y
(4.86)
X2
1-r
3
+ P 3 Y3
Z3
so that
X
Y
Z
2
3
3
+(l-p 3)Y 3
l-r
l-r
3
3 -P 3
as
i-r 3
Y -+ _ - -3
'P
- 3
-
(4.88)
The limiting behavior of i(U) for case 1 of Section 4.3 is analyzed
here as an example.
k
A complete list of the elements of --
appears in Appendix C.
k
1,..., 12,
-720, and both these parameters occur in the
In case 1, X1 + 0, X2
numerators of all
are both 1.
(s,U).
The lowest powers that X 1 and X 2 are raised to
In this case, Y1
+
(or alternately, Z 1
X
+
).
Parameters
Y1 and Z1 do not occur in every denominator, but they do appear in some
numerators.
The highest power of Y1 is the first.
Finally, Z3 + O occurs
in some denominators; however, those elements of 6(U). in whose denominators
Z 3 occurs all have high powers of X 2 in their numerators (see Appendix B),
and the ratio X 2 /Z 3 approaches the constant given in (4.87).
Thus,
1 is obtained by dividing _(U) by X X2Y 1 and taking the limit determined
for case 1.
Using the notation
(k)
(k)
(X1U
(k)
(k)
, Yk
(k)
),
(k)
Y..,
,
=
12
(4.89)
(where again k stands for the limiting case number), limiting expressions
are found as follows.
From Appendix B,
(l-rl
(r
(O,O,O,l,l,U) =
3 - rr 3
- rlP3
XlX2Y1Y2
(4.90)
r 1P3
Dividing by XlX 2 Y1 and taking the limit,
(l-r)
(,0,0,0,1,1)
where Y
=
2
r1 P3
(rl+ r 3 -
3
(4.91)
rlP3 )Y
is found by substituting (4.74) into (4.10).
Similarly,
(l-rl
-
01,,1,l,0,U) =
which, when divided
E1(' 0
1 ,0 ' 1
, ,)
Xrl
(4.92)
by X1 X 2 Y 1, yields the scaled limit
=
(1-rl(
r
1)
Y2 )(4.93)
Another element of _(U) is
(l,l,0,0,l,U) = X1X2Y 3
123~~~~~~~~~~~(.4
(4.94)
-73-
Rere,
Y1 does not appear in the numerator, and Y3 approaches a finite
limit.
Thus,
The
(4.95)
0
E1(1,1,0,0,1)
(') expressions for higher storage levels involve higher powers
of X..
For example, from equation (3.22),
2 3
5(2,3,1,0,1,U) = X X2Y1Y3
(4.96)
After equation (4.96) is divided by X 1 X2 Y 1 , the right hand side becomes
X X2Y3 and since X1 + 0, X 2
+
0, and Y3 is finite,
(4.97)
E1(2,3,1,0,1) = 0
In fact, it is easy to verify that internal state
always yield zero limits.
But the limits have even more structure than
that: the regions that yield nonzero
given in Figure 4.3
.(-) expressions
k(s) on the (nl, n2) plane are
If both X 1 and X2 go to zero or infinity, the non-
zero Ek(S) appear only in the corners corresponding to small and large
ni, respectively.
On the other hand, if one of either X 1 or X2 approaches
a constant and the other tends towards zero or infinity, nonzero
may be found along edges of the (nl,n 2 ) plane.
the limiting vectors E
k(S)
A consequence is that
contain large numbers of zero elements.
In
addition to decreasing memory requirements, this causes some elements
of B matrix to vanish and thereby makes it somewhat better behaved.
-74-
N2
ocases
cases 7&8
-
N10
nc
Figure 4.3.
IIk
Non-zero
[ks)
regions for k=l ,,
. ,12
5.
DISCUSSION OF METHOD AND RESULTS
The algorithm for studying the unreliable transfer line with buffer
storages is now complete.
2.
A Markov chain model is formulated in Chapter
In Chapter 3, state transition equations are analyzed; the form of
expressions for the probabilities of internal states is guessed and the
remaining expressions are derived.
Coefficients are obtained by solving
a reduced-order system of equations, as discussed in Chapter 4.
Numerical problems inherent in finite precision digital computers
give rise to serious difficulties in the implementation of this algorithm.
Furthermore, memory requirements are considerable, since the size of the
B matrix increases with storage capacities.
discussed in Section 5.1.
given in Section 5.2.
These difficulties are
A qualitative discussion of the solution is
Finally, conclusions and a discussion of directions
for future research appear in Chapter 6.
5.1
Solution of Reduced-Order Systems: Memory Requirements and
Numerical Difficulties
As discussed in Section 3.4, the reduced-order system of equations
(3.47) is obtained by deleting all the rows that are identically zero
in the matrix (T-I)B.
Since only a small number of E(-)
expressions
do not satisfy all transition equations, the dimensions of B (given
by equation (4.:26)) are considerably smaller than those of T.
It is shown in Section 2.4 that equations (2.14) and
gether uniquely determine the solution vector p.
(2.19) to-
It may be concluded
that for the present Markov chain, the matrix (T-I) has a nullity of
1.
Furthermore, it is proved in Section 4.2 that the nullity of 1 per-
sists in B, if {UJ, j
l,..,2
is chosen so that. 2 has rank Q.
The matrix equation
(5.1)
BC - O
may be solved by performing a singular value decomposition on B.
a detailed review, see Golub [1969], Golub and Kahan [1965].)
p is normalized so as to satisfy (2.14).
-75-
(For
The vector
-76-
The singular values are the non-negative square roots of the eigenvalues v. of BTB, i.e. of a. which satisfy
B BC. = a. C..
-
(5.2)
1-31
The vectors C. are the singular vectors (which are the eigenvectors of
T
1
B B) corresponding to these singular values.
If B has a nullity of
1, then exactly one eigenvalue of B is zero, so that if C
is the cor-
responding eigenvector,
(5.3)
BC=
-k
and thus
T
(5.4)
B BC = 0
k
That is, 0 is a singular value of B and Ck
is the corresponding singular
vector.
Because of finite precision, however, ak may be very small (e.g.,
below the range of machine precision) but not exactly zero.
If there
is a significant gap between the smallest and next smallest singular
values, then the smallest singular value may safely be assumed to be an
approximation to the zero singular value.
Its corresponding singular
vector is then approximately the solution of (5.1).
If a gap separates
more than one small singular value from the rest, the singular vectors
corresponding to all small singular values span the solution space, i.e.,
numerically, the singular vectors corresponding to each of these singular
values, or any linear combination of these, are legitimate solutions
of (5.1).
Since the nullity of (T-I) is 1, however, a unique nontrivial
solution p exists.
The singular vector which yields the correct solution
may be found by scanning over all those that correspond to small singular
values and choosing the one which gives a distinctly non-zero p.
Other
singular vectors yield nearly zero p vectors (before normalizing according
to (2.14)).
If, however, the singular value decomposition does not exhibit a
gap, no information on rank is available.
Computer implementation of
-77the algorithm described in this report fails to yield a gap when applied
to some lines with large storage capacities.
In such cases, the method
Ultimately, the measure of
discussed here fails to give good results.
the accuracy of any computer run is given by the error p -Tp.
The solution is found to be generally sensitive to numerical errors,
so that IBM extended precision arithmetic
(32 decimal digits) has been used
for lines with both storage capacities greater than 10.
The accuracy
obtained depends on the r i and Pi parameters as well as the storage sizes.
-3
between the smallest two singular values has been obtained
A gap of 10
for a case with storages of 15 and 16.
Pomerance 11979] used Honeywell
double precision (18 digits) and failed to obtain accurate results for
Extended precision arithmetic in-
storage capacity more than N1 = N 2 = 9.
creases both computation time and memory requirements by close to a factor
of 2 over double precision (16 digit).
This difficulty is one of the main
limitations to efficient use of the methods presented here.
An additional difficulty of as great significance is the large memory
Although relatively efficient storage techniques may be used;
requirement.
memory requirements remain high due to the size of the B matrix.
Although
values may be stored on slow memory (e.g., disk) or computed as needed, this
would certainly be at the expense of speed and may render the method prohibitively expensive.
5.2
Qualitative Discussion of the Solution
In this section is a short discussion of qualitative aspects of the
form of the solution outlined in the present report.
This section does not
describe those physical attributes, such as efficiency and average in-process
inventory, which are independent of the solution method.
These are investi-
gated qualitatively in Schick and Gershwin 11978] and Pomerance 11979].
5.2.1
Magnitudes of · Expressions
Examination of the values of probabilities p(n1 , n2 , al' 52' a3)
reveals some.distinctive characteristics.
Internal state probabilities
are generally smaller than certain edge probabilities, and these edge
-78states are less probable than some of the corner states.
The smallest
edge and corner state probabilities are on the same order as internal
probabilities.
This is not merely a result of the solution method described here.
If it were, this procedure would not be valid.
It can be observed in the
solution displayed in Gershwin and Schick [1977] which was calculated by
iterated matrix multiplication.
(See Schick and Gershwin [1978].)
The parameters of that case are N 1 = N
1
2'
P3 = 0.05, rl = 0.20, r2 = 0.20, r3 = 0.15.
-3
-4
are on the order of 10
and 10
.
10, P
= 0.10, P
1
0 05,
2
The internal probabilities
Some edge states (which appear in
Tables 3.2 and 3.3) have probabilities in that range, but others (such
as (l,n2 ,1,1,0), n2= 2,...,N2-1=9 have probabilities in the neighborhood
-2
of 10 . Some corner probabilities (in Tables 3.2 and 3.3) are as
small as internal probabilities; others (such as p(0,l,0,l,0)) are comparable to edge probabilities; still others are larger than 10-1 .
This may be because the p.i and ri values are such that when the
system leaves an edge or corner due to a change in a machine state,
it is unlikely that another change in machine state will occur before
another edge or corner is reached.
Consequently the system tends to
move through internal states until edges are reached, and along edges
until corners are reached.
To insure this, it is assumed, in the
discussion that follows, that Pi = Pi6 and ri = ri6, where pi and r i
are 0(1) and 6 is 0(1/N) where N = max N i.
i
The U(s,U) expressions found here tend to agree with these
observations on p(s).
Of course, some boundary state expressions agree
with these observations because they were deliberately chosen to
agree.
(See Section 3.3.3.)
Others, however, are chosen to satisfy
transition equations.
Certain edge expressions in Appendix B (such as
(l,n2,l,,1,0,U),
n2= 2,...,N2-1) are clearly 0(6-1 ) times an internal expression.
It
is more difficult to assess the magnitudes of such edge expressions
as
(O0,n2,0,1,0,U),
n2 = 2,...,N2-1.
contains a difference of two terms.
This is because the numerator
However, when U is on the closed
curve in Figure 4.1 (curve 7), then X.Zi. - 1 is also 0(6).
1
1= 0(6) and (1-r
(See Sections 4.1 and 4.3.)
i
+ PiYi) - 1 =
Therefore the
-79difference is 0(6) and the expression is 0(
1
) times an internal expression.
Similarly, some corner expressions, such as E(0,0,0,l,l,U) are
-2
0(6
) times on internal expression.
5.2.2
Valuesoff U., j-1,.,$
Several numerical experiments were performed concerning the distribution of U.= (X j
equations.
X 2j,
Ylj,
Y2j, Y3j)
solutions of the parametric
It was observed that the quality of the numerical solution
obtained varied with different choices of the parameter sets Uj, j= 1,...,
For example, a simulation is described in Gershwin and Schick [19771
from which values for X1 , X2 , Y1 , Y2, and Y3 were estimated under the
(incorrect) assumption that V'= 1 in (3.41).
The estimates and sample variances indicated values for X 1 and X 2
near curve 7 in Figure 4.1.
An experiment was therefore performed in
which the algorithm described here was run with all U.
so that Xlj and X2j
parameters chosen
fell on this curve.
The results were surprisingly bad: the numerical rank of the KxX
matrix B was roughly L/2 rather than the correct value Z-1.
That is,
roughly Z/2 of the singular values of B were indistinguishable from zero.
This may be attributed to the fact that all the Xlj and X2j were close
to one another, and to the block Vandermonde structure of B described in
Section 4.2.
By contrast, relatively good results have been obtained when none
of the points were chosen on curve 7.
That is, when points are dis-
tributed relatively uniformly on the other six curves in Figure 4.1, the
B matrix has unambiguously had a rank of k-1 for the same storage
capacities.
The smallest singular value has been indistinguishable
from zero (i.e., below the range of machine precision) and there has
been a substantial gap between the smallest and second smallest
singular values, with the second smallest singular value being clearly
non-zero.
P
Also, the vector p-Tp has been substantially smaller than
.
-80-
The best results have been obtained when the points are relatively
evenly distributed on these six curves.
Using the extreme points
(limiting U and r ) discussed in Section 4.3 and 4.4 also slightly im-Jk
proves the solution. Using only extreme and curve 7 points, however,
is not better than only curve 7 points.
The question of how to best choose the Uj points is not well
understood.
Considerable freedom is available for choosing these
points, and it is clear that how they are chosen makes a difference.
Further study is required.
6.
CONCLUSIONS AND AREAS OF FUTURE RESEARCH
This report describes a method that may be useful for solving a
wide variety of Markov chains and processes, and which has been applied
to the analysis of a three-machine, two-storage transfer line.
This
application illustrates that the method requires considerable involvement with the system under investigation.
It is not, currently, a
general technique which can be mechanically applied.
The complete program written for the transfer line is limited to
systems whose storages hold about 20 workpieces each.
It is limited
by both memory requirements and, more acutely, computer precision.
Further work is needed for the program to be applied to large storages,
and for the method to be usefully extended to longer lines.
In this
section, some research areas are described which may help reduce these
difficulties.
6.1
Different Boundary Expressions
When the E(s,U) expressions in Appendix B were obtained by the
method described in Section 3.3.4, it was not fully understood, at
first, that some transition equations would not be satisfied by these
expressions.
Also, the consequences for the size of the B matrix were
not appreciated.
As a result, no effort was made to reduce the number
of such equations.
= AN
1
As pointed out in Section 4.2, there are
+ A2N2 + C
(6.1)
unsatisfied equations, where Al = A 2 = 4 and C = -10.
A proposed research task is to try to treat the boundary equations
in a different sequence.
Edge states should be fully analyzed before
any consideration is given to corner states.
The intention is to reduce
A1 and A2, possibly at the expense of increasing C.
This is a cost
that should be willingly paid since it reduces the growth of Z,
size of theB matrix, as N1 and N2 are increased.
-81-
the
-82-
6.2
j
Choices of U.,
J
= 1,...,
The set {U1 , U 2 ,..., UZ }
.
is not determined.
Each point is required
to fall on the manifold determined by the parametric equations, but
there are an infinite number of such points, and only k are required.
Once they are chosen, {C1, C2,...,
Cj
p(s)
C } is determined and
(6.2)
S,Uj)
j=l
can be calculated all states s (subject to normalization).
If for a
given problem (i.e., a given set of Pi, ri, and N.), two different sets
of U. are considered, two different sets of C. will result, but both
J
I
will produce the same p(s) for all states s.
On
the other hand, it
is noted in Section 5.2.2 that the choice
of the Uj makes a difference in the numerical behavior of the solution.
Thus, there must be a best way of choosing these points.
For example, it may be useful to choose U.
some odd state s.
(See equation (3.25.)
so that g(s,U)
= 0 for
This will cause many of
the elements in column j of the B matrix to vanish, if s is an edge
state.
Another possible approach comes from the observation that if
U(s,U) satisfies all transition equations other than the odd equations,
then so does
m
d (s,U)
i= 1,2,
m > 0
(6.3)
m > 0
(6.4)
and
m
d (s,U)
dym
i = 1,2,3
1
where full derivatives are taken, i.e. where relations (4.1) and (4.2)
are taken into account.
This is because if both U and U+ AU satisfy
-83(4.1) and (4.2), then both
(s,U) and
odd transition equations.
(s, U+ AU) satisfy all but the
The same is therefore true of
0
AX, I(s, U+ AU)
(s,U)] ;
-
i=1,2
(6.5)
i =1,2,3
(6.6)
and
A1
[(s(s, U+ AU)
-](s,U)],
where AU = (AX1 , AX2, AY,
1
AY 2, AY3 ), and it remains true as AU +
.
By
the same reasoning, higher derivatives also satisfy all except the odd
transition equations.
Consequently, derivatives (6.3) and (6.4) can be used to generate
the probability vector:
2 .
p
s)
m.
E
2
i=l
3
E+
i=l
where
ij
j=l
=O
:
m:
' 1
C·
e-
=
j=1
as before, the C..
m di
m
(s,U)
(6.7)
1
and C!.
ivij c
(sI)=
(U= U
ECE
m E ij
a3m
coefficients are obtained by solving
a suitable set of equations
BC = 0
(6.8)
p(s) = 1
(6.9)
all s
The reason for investigating this is
that that resulting B matrix no
longer has the block Vandermonde structure of Section 4.2.
its
blocks have columns which are combinations of
In fact,
-84-
1
0
0
X.
1
0
X.j
2X.
Xij
:2·
N.-4'
X
N -5
((N1
I
i]
4
ijJX
,
..
N. -6
) )
1(N1
5)
(-4)X
These vectors are linearly independent and not numerically close to
dependent.
6.3
Alternative Models
The transfer line problem is interesting and difficult because
of the boundary behavior.
It may be possible to make it less difficult
but no less interesting by slightly modifying the boundary.
New models
should be investigated which have the qualitatively important features
of transfer lines, but whose boundaries are easier to deal with.
This
again is intended to have the effect of reducing A1 and A2 of equation
(6.1).
One possibility is the exponential service time model described
(for a two-stage line) by Gershwin and Berman [1978].
This appears
to be a likely candidate because the boundary seems to involve only
two storage levels for each storage (n. = 0 and N.) rather than four
(ni = 0,1,Ni-l, and Ni) as in the deterministic service time model.
1
1
1
(6.10)
APPENDIX A
A Set of State Transition Equations for N1 = N 2 = 5
The FORMAC program described in Schick and Gershwin 119783
was used
to produce the complete set of state transition equations for recurrent
states for a transfer line with storage capacities N 1 = N 2 = 5.
It may be noted that this case is the smallest (in terms of storage
capacity) for which the upper and lower boundaries and their immediately
adjacent internal states are completely distinct.
The transition equations thus obtained appear in the pages that
follow.
-85-
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-3
, 1,0.1,
----------------------------4**FQUATIO3
NO.-------21^----------I----------------ZERO
- 1,
P.()2
1, I 2, 0,,PIP3(
1,
P.(2 ), I
1, -p
, 0 .I)P.
) P2 P1 C2,.
3 +
-I,
P.1
,
1,
-,
---------------------------w---------------------------------41)C~~~~~~~~~~~~~~~~~~~~
-P3+
0F., + I )?~
I
,2 3 +
I. +,
) I
2(
) )[-1P-.
F
1, 3,0,
82 83 + P.(
- P2 + I I
o
2
I1
0)
-----
,)PZ*
P0 · (
1
0,
:,
19
1, 1 ) P2 , {. - P24-1
P
O
I
1
2(
ZER3
li 2,1,32_
0, ,0I'0,0I
LI, 3
+ ) E3 * C
2., -1 , R31,. tI+)I P)2 PIP 3 ( 2, .
, - PI +4 1P. )( ?.C
*-*PQUATION
-
2
)P
-: C, 1)
2,0 1,
i,
1, 0, 0,
----
1)+ P.(
11
1i
-
1,
) 1)
* P.( 1, 11,~0,
-P
4 t P1__
2,c 30.
I, 1,
1 1, 1,) 1·_"r
( ),
- 0 )1)32 atP. 3x
***L2UATIoN
18w
----
1
-.P
P3
, 0+0,)[ I1
I )1, P..
.
· 1,t[
,
,O
(.
I
8
-1. 2·0 I)
2*IIt
..
l
0
…3
…2
…
P.2 n3I
Al
I2*)
,
P * 1P D
-
2
-R
P1
- Pi2
·
2
) [
t
t
-88*,4.;:I.TIO!I
a
RJ
1,
1
29
.'0.
7.~;~:) -- - ,'.:
l,
,
1,
1,
- pi
3
**$
1 ) + p.[ 1,
1 ) P.
1,
3,
3,
O,
0,
ro2 P 3 L I
0, I ) 1-2 RI
3
P
0,
- P3
:, I, ,
)
, 4,
', 1
1 ) P2
81
I
, 1 ) iii
I
- P2
P
, 1
3.
, I
4 - 23 4.
1 I
----;.---- ---- --- --- -;----- --- -------;- ---------- ' ; -------- ------- ----- ----- --- ------- --- ---- -------- --- ----- ---- 1 )i'.l1,
3,3,
.1,1)
I1
+ I ) P. ( 1, 3, 1, 1,
..........................
***LQ39I tON 3.
ER3
- .
, , ,
P. ( 1,
1,
4,
-
2
I2
- P11
PI+)
I ) P.,,1,
3.
1,
1
) 343#
,
-P2 21)
-
PI1)4
I
P3
I )
,
1, 3 ) ,2
33
P.
)
*!o
1 ,
- C -I3
1
1,
,
1, 1,
P+ Pt
1'3RI
*1
P3 +
) P.I 1,
l +1 }
-
4,
0, 1.
P.
1,
4. ),
3 ) P2 *
-
1, I ) P2 P3 .
2) + I
- 13
3 * 1
-
1I
RI 4.
-
1 ) ?. 4 1, *4, 3, 3, 0
...................--1 )~~~~~~~~~3
:'**L. 3X(.
C0.
31
.***
lEr,o
- ?.( 1, ', O, I, 0) . a.( 2, 3 , 1,
1 ) P1 12 73
:. - R1 * 1 ) P.: 2, 3, O, O, I ) 112 P3 * [ - p2
.I )
.......................................................................................................................
P. ( 2,3,
1, 1, I)
1P3](
-P2
+ I ) ( -1.1)?.(2,3,0,1,1)P3*(
- R3.1)P.(
2.3,1,
, 0O)P1
........................................................................................................................
P2 * 4 - E
.1
- 1
1 _) P.I 2, 3, 0, O, 0') 12 * 4 - P2 * 1 ) ( - R3 * 1 ) P.( 2, 3, 1, 1,. 0)
P1 4 ( - P2
+ 1)
: -1341
I)
......................................................
*F.IjATIN
LZEO =
· 0 )
- P. 1,
P
- P1t1)P.4:2,
!13.
3+(
3.0,
1, 0)
32
.4$0
4, 3, 1, 1 ) * P.: 2,
- 22
)
.(
4,
2, 4,',
1,
1,
0,
0 ) P1.3
+( -R1
1)
P. ( 2, 4, 3, 1, 0 )
3
1
-2
+
) (-P31).
.......................................................................................................................
* 1 ) '. 4 2, 4, 0, 1, 1 )
4***E..23AXTF
I'NO.
LERO
, 1)
-P.(
1,
4,
1,
1,
P3 Pi.
(
- F.3+1)
i.C.,3,0,
32Re(
1
1 ) P.(
+ I ) P.1
i,
1,
***
1)
2 P3 aR · P. 43,
4,
33
·
P.( 1, 3.0, 0,
0
t * ( '-
3 ) PI ?2
P ( 2.,
,3)3.
) -3
5,
)
0,
2,
3,
0,
0 ) g2
1, 0 ) R3*(
P1
I -P2
1, 1
) P.{
4,
0,
1) -1
) P3 R1
:
4O, 0,I,o1,
3 + P.(
-P2I
(
I,
-
1
R1 +
P3
- P2
31
1, 5,
)
-2
I ) P.
1, 3,
-P2E41)
1,1
-I33
1 ) P.
1, 3,
,1,1, 0.) [,I +
-P2 + 1 )
- P1 +
P. 1,
3, 1, 1, I ) P3'+ (P2 + I )
- P1
1 )
3
.-.......-- - -- -- -- -- --- --- -- --- --- --- - - -- . . -- -------------- --- --- -- - -- .- ......
.. .
------ ----.......................
.i I1
P.
1, 3, 1, 1, 0I
..........................
***EQ~UATION
0D.
34
***
ZEFO = - P.: 1. 4, 1, 1, I ) + P.: 1, 4, 0, 0, 0 ) i2 F3 Il + P.( 0, 5, 0, 1, 0_)
3 R1 + [ - P2 · 1 ) P.[ 1, 4, 0. 1
, .1)R3
- 4
I1.4 '
-P2+
41,5,
,)+
***EQUATIOU
-P3
P
-P2+
+ I )3 P.
,1,
-3
3,
:2,
*
1,
1, 1 )
C
-
)I
1, 3 , 1 ) PI
2+
1 +
1,
-
. I
1,
4 + P3
- +
:-P3
P2
-1
P.42,
-
--------------------------------------------------------- I2 + 1 ) P. 411,
3, A,1
1 ) Pi P3
-
[R3
, 1,
-2PI?
' (
f1
, 0,
O,
3 )
-
+ 1
*
1 I
2+
-
I ) P3
1
2,
2
-
1 '(
I 1 3 P.(
2.
4,
0, , 0 -3 0 ) R2 2.
-1
I
+ I ) P.4
0.
1, S, 0,
1,
4
0
, 1.
-,
1,
I ) P2 +
1,
4-
R'..
t,
0, 0
P2 + I )
) 32
0,
-P3.
P-------------------)[ - ,1,
I ~,, I
112+[-1
--------------------------
I
1
(
1 4 1
-
P_
0)
)-
( )2 4
-2.IR )
[4
0, 1 ) P1
4
2) I 4
3-2
-11R2
I PI1 .(
1I )
- R1 + 1 3..
1,, 0,
-..
0,0
?.
4. i
2,
0 ) B3 31 -4
)P1, .
1 - )
1,
I.
P3
I
P.
+-- , --------O,
1I, ) -R.
P3 + I)
P.
1
1[
1,1,.
1.
1,
0I, 1,,
O,0
,
) e
-------12,
) P2 RI
1 +
.
-,
1, 1
---------------1,
) P2..
--------------------------------------------------------------------***.EQATIO3
NO.
39
ZEIO3
3. 3+2
,) 1, 1,
+ 4. 0.
- P20, +I 1 ) )P3P.
2,4 I1,- P+
1. 0, ) 3
P- R33 + 41 ) - P.[
311
P.
2,- -, .P.[1, 2, 0,1 .1,
)
: - P.t2 *, 1 .1.
- F )*P21
1 ) P.P32.,
---40---- - P---4 14 )-P
-------. 1) 1 3243-- +
2 - 1-, -1 -- . - 1 -)2,PI- )-- P.1- -- - ,-- 0,- 0,
-- 0
- -- - 4-- - ----. 2 -12---0, -1 - - --2--1,***
0,, EQUATIO'
1
)
N3. 1
ZE1RO= - P.4'2,
, 1, 1 ) +41P.4 3,
2
, 1, 1, 3 .**
) PtI3233+
- 1R2 + I 3 P.( 2,,
1, . 0, 0)
P2 33 + 2
3, 1
P. 1 2,
2, O,
0 ) P2 13 + :
Fl2 1 )
: -, 11I + I ) P.
2, 2, 0, O, -0 ) F;3 .+ [ - P3 + I ) P.{ 2, 2.
1,
I
.......................................................................................................................
t'1 + [ - F2 + I ) , - P 3 + I ) P. [ 2, 2, 1, O, I ) P1I + t - P 3 · I ) [ - El · I ) P. t 2, 2,, 0, 14, I I P2
+I
. 1I.1)
1
P-).
0.01)
-----------------------------------------------------3*:
-P
1 +) P.
1,0.1,
1, 1 ) , P2+
-p
--------------------------------P. ' 3,
)1,
1, 0,
,+ I )) P11
314.:-P4.
1,
+
~
·
I ) P.(
P.4 1, 4P 0. 1,
1. 0, 1,
1
4q,
3I
+
-3
-01' (
P
***
( 1, 2.
1 0,
)
1 +
+ 1 ) P. 12,0.
-3,2
.1 + I ) P.4 2,
-
2
1,
***
, 1,
-41+1
j P.2,2,
1.,
1,1,0)
3
.....................................................................................................................
0)
R3 + I ) P.
1, 1 ) P2 Pi +
)PP.{ I
- P2 · 1 ) 4
+ 1 ) P.
1, 0 ) Pi +
,0,1,
43 -
1
2 + I 2) ,P.
1)I-2.1) +
* 4
1
+ 1)
38
3I, I1
1, 1,
2 P. ( 24,
1)
I)
4..
37
1 ) +C
3 + 1 ) P.4 2,
***EQUATIO
NO
3.
ZERO = - P. : 2, 0.
----0 ) P2
+
-31*F
- P2
+
1., 1)
- P3 + 1)
- P2 + 1 ) P.!
- P2
***EQUATION NO. '
Z"_: = - P.: 2, 0,0,0,
)3
2
1 )I
1,
4,
0 ) P1E 1.
+
0) +
) 03+
1,
4,
- P2 + I ) P. ( 2,
+1)
1, 0)
0,
1
36
, 1,
, 5,
1,
)1 P.
.*
+ I -
:
NO.
- P., 1. 5
.41,
35
*
,1
1)
I ) p. 41,
+ 4,
P3
- ·
) P.2, 2,, 4, 1.0
1,1,) 1,
8Z£O =
P3 1+)
...
0, 1, 0)
+1
4
:
P-1
***FQUATIO; NO.
ZERO
- P. 1, 5,
)P3 3 +
1)
1 )+
P2
-
-23.1)4
~ ~ ~ ~ ~ ~ ~ ~4 ~ ~ ,O ~ ,O~ ~ ~2*[-P ~ ~ ~ ~
2, I
-
, I 0
J .2
i t
- i12
-31
I
-...
-89-
~~e
=
-
lO~
l
NO
P.:
2,
,eHJ:l&
Zao
( -
1,
H3
I
1,
1-
) P
I. , 0
I .( ,I
'.1,2,
*
+
- 21
· tF4QiAT'
i
EP
EO
I
,
Ji,
' 1 )
N
- P.(
2,
:
·
3,
·
1 I
1,
-
Pi
P3 F1
p2
+ I
)
P.'.:
4
C - 22
1 t ,
,1
+I
, I,
)
) P.:
P2
P3
1,
',
1,
+ {,
-
,I
)
C - 22
P3 21 *
1)
P?1 + I
[
-
a.3
·
I
J
1,
1,
3,
P1 I1- (
--- --
O,
**£Q]ATION
3,
1,
2-
=O
P. .1,
(
1,
4,
1,
1
3 I
, +
1,
P3
)( -
P3
,
,3
,
----
0,
I
R2
R3
RI
1,,
- - ---
O,
+ I ) P. 1,
O,
I ))
. 2.
0.
+
·
P.:
2,
0,
0,
0,
I1) )-.(2PI
0-PI)+
- - - - - - --
P.
2,
2
0 1,
o
1
) 82
!)
- --
0.
0
P IP
I
R1
{
R 3 R1I +
) P.(
1, 1 2,
)
-
,
,
-
2 3·
1)
1,
,
,
-
- P3+'
I
)
C 1,
)
2,
0
.1
1
3,
1,
1,
J
P2
P1
3
* (
- R1 + 1 ) P.: 2,
2,
* I
) P1
1,
}O, ,
1,
1
1,
1 ) P
-
I *
.
1, 1,
)(
) P.
{ -
t3 2
1I)
--- ---
( 2,
1
2
--- ----
)
--- ----
+ I
3. 2
,
,
0,
.
ZI
1)
2,
+
-
- R2
1 ) P. ( 2,
O,
0,
3
P3
'P.C
1 )
1,
1,
13, 21,
I
1 ) P3 +
) :
- E1
I)
-R1
3,
3,
0.
1,
1
+
I
1)
-22
1C
-
) P.
, 2.
0.
+I
I
2,
3,
) P.
951
+-P.C
R1I
3,
1,
0,
C -
3
· I
+ 1 ) P.,
P.2,
) a3
3
0)
P I3
) P.(
2,
,1,
3,
0,0
C - P3
+
- 1- R
+1
)
(
- P3
P3l1 *
P12
,
R
31,
+ (
2,
0,
3,
-
1,
I
), P
12
) P.[
3+
2
3,
1,
3,
) P
3
.2 ]
3, .PC
P.
*1)
12, 0., I,
I
1,2. 0,
) P3
- R3
0,1
) P3 P3
+'1
) P. C
*
, 2.
1,
1
2,
) ) 22 33
R2
,
2,
.
1, 0,
) P.
,,
1,
I2 ·
,
)
31, P2
32, 0,
+I
I
0t
) P.
) - 31
3
P3
13, 3,
+C
I
)
0.
0,
0 )
P3 E*3, I
P.
23
)2, P.C
1,
i
C
·
1,
1,
3,
I
}
2,.
)
1,
53
) * P. : 2,
- - - - - -- C
3,
1,
- P2
2 P3 E1
0 )
2
,
) P.C 2,
1
C
,
,
2,
) P.{1.
2 )
. 1 ) P3 R1
P+
P.
(.
- 13
11,I )
P2
.
*.
1, 1,
1)
P2 P
- -- -
* I
*
{1
O,
-
)~
- I~ ) P.[(
f [I + I
!N0.
2,
1,
R3
C - P2
--------------- - -3
- --
....
+ 1 ) P.
2,
- - - ---
3, 1.
.....
0
P2C
I
+
-C
-------- -----
P
.
1)
P
-----
,"3 * I
)
)
P.
3,
E 2,,
1,
- P1
13
O,
3,
0,,
1,,
1,
0
+ II
)
)
) Pi
P.(( 2,
2,
P.
2',
3,
O,,
1,
O,
*
R3
* I
4I,
O,
II
)
,1 *
}P3
(
El
#:
[(
I
-- a3
P13
-
I
I
0)
I
+ I
P,.(
2,
I...1
P1
""
C -, 11
21
-
)P. 1,
-----
3. 1.
-----
0.
1. 0)
O,
1,,
PI
C -382
P.
]3 ·
) P}
Pi1 P3
C - P3
El
1
1,
1,
1)
-
I.
----
1
1
1
,0
*
4R'
I13.
3)
22
.2
(
R2
0~, 0)
5llte
3,
0
)
2,
C
(
)2
"~
1,
4,
P.(
3
++
I1 )
}
I1
1.I
C - P3
~
.,,
P. ( 2,
-
)
- i.
P2
2
P2
P2
+ 1 )
P
.' 1,2
,
) ,R2
3,1
- -- - - - -- - - - - -- - - ------
E1
P3 +
+ ',
(
03
1 ) --
*
1, ,
2.
)~ P2
) ?2
P2
)
1
P32
, Pi1,
0
=
1)
) PI
1 I
), P., : -+C
,
2,
C -
3
3,,
eCEQSATION
2
32
3.1
3,,.
ZERO
1)
) P12
-
)) P.
2P
P. (
f 2,,
Ii
-;3
I
))t 1
P.: ( 2,
2,
P.
}
) P2
- Rt2
,
3,
+I
1)
4
C - 32
1,
1 ) P2
..........--
21.+
( - P.241
P.(
2, 2, 1,
3,1t
.0.0) ) P12 * ( - 33
* 1 ) C - 214 1)
P.C 2.
----------------------------------------P3
+1 )
1I ) )(( C-0r3
·4 1)
+
) 1i ) {
-- 31 '3
+1
I ) P.C2,-0,-1---P2
} P.(?.2
2,
2, 1 1,, 0,1,
1
------------------------------------------------- ***EQUATIOi
--------------"
-------4*
NO.
52
·
**EIQUATION NO.
53is
ZERO
- P
P.(
2,
32, 0,
1, 0,
0
P.CI 2,
2,
2
1, .3 )I
P
21 P3
R3 · C -2
1 )
) P.:
P.C 2,
2, $,
, 1,1, 3,
ZERO =
.: 2,
3/'
01 )
) * P.
3,
1,, 1,
! P2
PI
-R
+ I
-- - - - -- -'---------------------- - -- -- - -- - - - -- - ----- - - - - -- - - -- - -
* I
1
1, 10
0 ) R2
CO, -O',83 I +
) 1 82) P.
+
·
3,
+ 1 ) P..
10 ) ,2 h i
-------------
R2
----
)
+ 1 ) P.
21 + II)
-
R3
1,
P2 *3, 1) 2,
- P.'.:
)
+: I
31
1 1,1
***EQUATION N0.
ZERO=
- P. : 2,
3, 0,
1,
-- - -- - - - - --
* {
)- P2
---
*4*
1 ) + P.C
-
) P2
+ I
1,
1,
0
.............
1), 2 3 , , 1,
+ P.
+
.3I
1,
0,
.'),
0,
-
~'"
2,
3,
1;
...........
3,
p.t P. 2, 2, 3,
),
.-------- ,
- :2
1,)
-
+ I
O,
0.
P.C, 2.3,
-----------
2,
48*s
5
- +( 11)-
- P1
+ I
0 ) P2
P.:
,*EFQUATION N O.
52
ZERO =
- P., 2, 2,
1, 1,
1 ) · P,.
2, 23. , 0, 0)
P2 i3 PR +
- 22 + I ) P.C 2,
23,
, 1,
0 ) 33 Rt1
-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- - -- - -- -- -- -- -- -- -- -" --" --'- -- "
-- -" "-- "
--" ------------{
-P3+1)
P.,2,23,
. 0 1I)
i1P+
t-PI1+
I PP.C2.23.1.,
P.12,2,0.1,
1)I 2F.1R*
:
- R2 + 1)
a3 + (
P3
l
2
1. 1, O, f O)
22
?
3
- Pi3
- - -- -- - - - - - - - - -
P.(
--
--- -- -- - P
- -3 --- -t C ----- ----
-
.
2,
)
?344*' 1)
-
--- ----
2,
- P2 + 1 )
'
4
1 ) P2 El
+
.1.I
1,
R2
2,
I ) P2
47
*atEQUATION NO.
.C 2, 2,
ZERO
2,
1,
+Pit
P
22
---
1)I2
*
) +P.
R1]
- P.C 2,
2
3,
I2
P.3 + 1 ) P.
1, 1
N0.
RI~
,44
.
465**
+ ?.,
1.: )I
1+1)
- ----
R3
) P. ( 1,
1,
, 13,
48
1 )
1
=
+ :
12 23
0U&TION NO0.
- P.
2, 2,
-
P.:*
a2
+1.
?3
1,
1 )
*EQURTION
ZE:O
1 )
2 + 1
2, 1,
3R2
P1
1,
40644
2, 2,
- P3
*E
,
14.
1 )
ZER
) P2
2,
...................
3
1.,
no.
-P.
2,
2
45
---
----
.
2,0,
1, I.
1,
+ P.(
*.
3
I
- a2
. I ) '
- 23
---.-- -*1-- ) - : -I----.--
zEO =
2,
·
2,
P.
-)
P
---
1
~1P~S3.N0.
1 )
I
1
...
*4*E;']U;LTI0N
ZERO =
- p.
2,
1 }
92
-3
41
--- -2
-2
1 )-----
Pi'
1 )
P. [
4+1
2 , O , I , O)
I
- P3
1,
1
) P.C:.
--------------------------
P.
:-I
43
+ P - ( I,
I
140.
P. 44*4ia4'htLN
)2
2,
0, 1 13.1 ) P2
P.
)
.1
1)
.C221
------------------------------------------
P.[
1,
1,
,,, eO, 0
2,,,F2
.
P.
) ".:
,
0 ) F2
4*LuA!rI,)s:JO.
ZL,-,O =
- 1. ( 2,1
h3
,ee'
12
1,
{
0,,
0,,
P2
113 + (
- P I
I
)+
+ I
P.(
2',
.2
+ I
) P.[E
2,
4,
I)
PI
F
4,
O,
P2
+I
P1
k3
P'.; ( 2,
1.[
2
1,
I ) P'2
......................................................--
O,
'*I
0
)
)
P.C
;..3
2~, 4,
1,
P -;3 ·
O,,
0
}Plii~
I
)
P.{(
93
2,
[
i 1
I1,
I
t
I
I
!'2
-90-
~
P.
·
.
m
.1':UATTON
.r:3
=-
30.
2,
3,
~ ,,,,;''o.5i**
'1 , 01)
3, :·Z
2, 1,*(-i~~~3,))!
1, 1 )
21."
,
.
2 +
I1 ) ' - P3
- 23 .1)+ 1 :
I ,
s. PV.2
1 t,3
: - - 11
El
I I.e1
- H
21
41
I)
J ,I 2,13 I, ,~,
*1)-1, 3,I,:
-R I*~'Z
1&
) ~,,z
P.'v3 3,·. 2,t.Or0,
1 )
)I ?.:
) P. 3.
(, 3, , 2, ,0.
(,
4
3
1,0 )1 P2
)
( - P2
1)
(
4. , 2,
PJ
3
3,
3:, 1)
.2
P3
- R3 + 1 ) P2 3,
- P
1)
P. .13,
( - P2
4
2,
2 i, 1,
, ,,
Ii
0 ) p1
0 ) P1*(
- 12
#t**E.)
pilT
NO.
57
,
*4
Z
==O
P.{
, 1, 0,I
) + P.
1, 3,) 0,
+ I ) P.' + 1,
0, 0, 0 ) 233, 133,+,
1
I2
1, +
1,2,1 3,
(,e, 1,
-O,RI I )
,') 21
1P2' E23
P. C
( ,3,- .3,2+ O, pI
-3, P3
?2.: *3, 'I 3,- P3
),) : PI - F3F.I · ( 1 )- ?P2.' +3,1 -~
+
1)1, 0( ) - R.3
P3 *{ I )
P. ( +3, I ) 3, e.1,( 1,
I ) PI O,., I( ) - PiP2
3 1:
P3
-
1)
-
+
1
) P.
,3,
I)2
31, 1.
*
58
E*:QU.OA.3:NO.
ZE.C
P.: 1,2, 0 3,
) *-P.:f2 1,
, 0, 1, 1,13 )
'3 21 1,
?.: 1,- 3,- PI
) 1,
P2 0,
.1E.3· 1
++
+
1 22
?3
r 3, +
I
I ) P.,1 4 1,1 )3, 2 1, - 1,
72) P,
P'
224 ' **tQUATI:0
(
3,
,
3, :2
O,
(1 1,
0 I ) 21i P. *
3,.-Pi0, 0.,I 4 0
) ),3P.( 1,
3I ) 1P2 )
1,
I( ,-I
3,
598*
N3.
~EO=
- 2. 1 2,
3. 1, 1, 0 3 * P. i 2, 2. 3, 0. 1 ) i42 3Ipi *1 1 - P2 4 1 ) P. 1 2. 2, 0, 1, 1 3 23 21 * ( - 23 4 1 )
23.
+-P241
2,2.
1,3 0.O9)
ZE=
- P.
2, )
3,:-P141)P.:
1, 1, 1 ) i 2.' 2,
3, 1,1.,1)P34:
0, 0 ) ?2 23 1l ,-?l1+)- 22
+ 1 )-2341
P. 12, )P.1:2,
3, 0, 1, 2,
0 .
R1 + 22-P3
*-22
1)
13
lP1 .C2
2. 1, 3.
) 22
) 22 P.1~
*
-?2
*1 !3
3
41)
3
2, 2. 0, 1, 0 ) El . I · 2. 2. 4, 3.
.3:
(
,
2,
1,
I
)
P2
H
*
1
)
P.:
1
2,
+
,
3,
1
3
4
)
2
23
+
1,
P
I
1
P.(
)
2,
1
0
,
1
4
,
22
,
?12
*1)*L*EQUAIO
-2141
r)
(
TION NO.
Z!J.2 - - P.
2, 4, 1,
?.: 2,
2,
)
1, 1),
P(2,
3,
59
P. 2, 4, 3,
1,1t,1O)h3:P
0, I
P2 P3 PE *
2.1
2
+1
- ?2 + 1 ) (
- P3 + 1 ) P.:
+ 1,)
P2( I+
I1
......................................................
P.
2,
) ( - 1
3
31
LQUATION NO.
Et
IE.3 =-
~
- P31).)
0, 0 )
62
4,
I,
1,
)'
?.
23, 3,
1, 0,
2, 4,21.,
+ )1 ) P.P 2.
3.
*4*l
1)
P 12
3~).(2
P.'
2,
- P2 + 1)
0, 0 )
0,
PP3 +
1
2,
O, 1,
( -3+
,1
I )
1)
2
P.
,1B+.
P3 * ( -
-)
2,
2,1,
) (
1-
+I ) (
*r#*QUO7AION
ZERO = - P.
NO.
2, 4,
3,
- P3 +1
63
1
P.
1 3
1,
) P.(
2,
3,
3,
1.
4, 1.
0,
- P'2 + I ) P.
1 3
0,
P21
P2 P3
3,
32, 3,
- R1 * 1 3 P.: 3,
*
1,
0 )
32 R3 +
2 . NO.
1, ***EQUATION
1, 3.3
Z-R3 = - P.: 2, 4,
, 1,
,1)
2 + 1F3 )
+ I ) P-
65
0 ) + P.
-3, )
P13 1)
- .
(-LP2
,1,
23, 3
1
2
, O,
. 1)I
-
1)P.2.
66
**EQUTJATION N3.
ZEEO = - P., 2, 4, 1,
, 1
* P.
----------------------------------'---------------------------'
3,
- P3 +
)
2P +
, 3,
1,
I ) R2 83
5,o ) -,)3. P2
- -----------------2,
-
I
P31)
P.
P2
3R2P3 +
1.
I )) P. - 2
- --------------" - "'
(0
- 23 * 1 ) P.( 1
34,
, 1,
)
P2
1P34
3, -;213,1 + I 1.}:
, 1 P2
) P33 + I )
- P.
P1 I *
+3,. 3, (.1,..-I 1,)3 0
**3
1 ) 2PIR2 R3
- P2 + 1 ) P.
23,
3,
0D1, 0 ) R23 31
3
P.4,
I 1, ,P1, 0. I -),1P2+1) 2- P1
P3* 1 3 ) P.
P.(I, 2,)R
3. 3.
a 0.
.
1
,
( -
1I2 )
P. 0[.-P2 .4.
PI *
P.
2. 3
3 -e+ I)
+I1 ) 24
( -
-
3
,(.
1
I-
I
i1 3
S-~I
, 1
*4*
4,
2
231
1,, 5, 0 , *1,
P) 0 1
2
67
66**
+ P. [ - 24,
P2.4, 2,.3
4, R.3
1, El +1P.[.
3 1, P3
2 5,
O, 04, § 3, h221
0I
P2
.-
3,5,
34
P.R -----------------------
P2
RI )
2,
) 22
1
- P1
4 P3
.
- - P3
2, P4. (F.
- 22 O
3 ) 23 El 4
21 23.
I1 P
,1,
1
0P23
31 3 *IP.1 2, P 4, 1,1 0, 0 PI+
)1 1, P.1) 2.l21 4.
,* 1 3 )
F 2 *+
, 1 1) P4(
I,1 I,P.E,n] IP25+ _ 1,1,
+ I ) P.
2, 3, 1,
3 ) 2 +
I )I ) P. t 2, P.3,1, 1,1, 1, )4 ) 3, + ,
, P1 +
)3 I
P3
-2:-PI
23+ + I
[
R+
3 4.
+I )
NO.
·*EQU3ATION NO.
·***EQUATION
ZI.5
ZSl:O
= = - P -P. [ 2,
2, 5.
4,, 1,, 1 1,
1
0, 0 j R2
1.
*e
1, 4,
0,I +
0. 0)
+
- P14 01 ) I)*1R2
P..
- R<I I+ )P..'
I ) P. 3,( 3,3, 3, O, O,O,
1, 0 ) *
P.- +
3 31
3
1 +P.1 )(.-R
, 4,
a2
e * i) ( R3
- R23
+I ) (
3.
()
P.(P2
3,R '4,+33 ,1,
624
P2.
0
24,
1.
) ( - a3
I ) P+ 2
41)242,
.- ,
P. ( 2,
12,
1
P2 ,1P1,1)P-32
) P.( 3,4.
-- - - -- - - -- - - - -- - - -- - - -'"-- '- - "
-- -"- "-- "
- ' - -" -l
- c -'--------.
1 ) P. ( 2., 4,
3 ) ( - P23 + I ) ( - R3
0 )
+ ( ,- 2
1,P.(3,2
4 ***EQUATION
1 ) P. ( 3, 4.
NO. 0. 1,
Z722 = - P.
2, 4, 1,
, I )
02
- P31 .I ) P.
2,
?.
2, 3, 1, 1, 1 ) B2 P3 4- ' - 22 * 1 )
- 13 + I ) P.: 2, 3, 0, 1. 1 ) E3 +
----------------------------------------------,,,,,--,,,--,,--------------------------------,,,,,,,,,,,------------R32
( - 23 , 1 ) C 1 + 1 ) P., 32, 3. 0 1 0, 0 ) 3 +
- P2 * I ) 1 - P3 + I ) P.
2,.
3 3. 1. 1.
1
3 * I1)
+
3 1RI )
04,1 I1,-I 02.1) R1
-
2
4.
P2 · I P.1
}P. 3, f 2,
4.
0.
I
-- 11.
P4,
P2)
0,
4.
1. I
1
0.
· 3 ) R3 El + '. - P2 * I ) ;, - P3 + I ) P. [ 2, 4s, P., 1, I ) EI1 + [1 - Pi + I ) P. [ 2, 4, 1, 04, 0 ) ir2. a3 + t
- PI
P.-, 4-2- 3. 1---2,3- I ,1)1 12
*(1, - 33
1)
P.( 3. 4,1I,
P3
)I -0..0·
1 -- ) 21,
( 0,OJ-- P13-13
+P. -2-3--1-)
) , 4-- 13,, --- 1--1-)-P.
, P1.
)0
- 1- -P.-P3
2. 5,
' i )P1
1
I ) P.:. 1, 5, I, _,I _O) _R3_-+ 1:
P2 + I ) [ - Pi)
+
I ) P.:' 2, 4, 1, I1, 0 ) R3 · [ - P2 4. I ) ( - PI · I ) [ - P3
1,
1,
I)._.
·eeE~J&TION NO.
ZERD =
- P. :1 2, 5,
0,
1,, 0
+ I ) P. ( 2,
)P3
{2,
· 0
4,
+ -!33
5,
1,
1,
+ I ) P. [ 3,
0 ) PI 4.(
} +
;, - ?2
+
I ) [.
)
4,
67}
+ - P2
1,
O,
+
I ) P.:.
0 ) Pi R2
2 + I ) (-R3
-
P3+ I ) I
+
3,
~
,1,
14, I ) PI ,°3 ·
ra3 + I )[-RI
4. I ) P. ( 3,
a,
1,
1,
-t2
+ I):-
· I ) P. [ 3,
0 ) PI
4,
R3 + I
- El + I ) P. ( 3, {4, O, 1, 0 )
...................................................................- -- -
RI + I ) P.-
3,, I%, 0,,1 I,1
O, O, 0 ) R2 · t
- B3 4. I ) P.
R
· -I I } P'. [1 2,
S,, O, I
-91U ATC3
-,
i
).
48
- p.( 2, 5,
.,
31
+I )
.'3)
I
..(
a*"Ft;UATioNi
)
3,
0) }
0,
:
1,I,
NO.
0,
O,
3, ·1 )
I ) P1I
(
I.[ I
2,
3,
93
) P1
0,
I1,1 ) P3
- E3 ·
L1 ·
I ),,P.(0, 2.
- P3
1 )
-
3,
,
) P.[ 2,
0 ) R
0,0 , 0 ) R
a,
- P2
(
-PI*1)
8, 1, 1. 0
.
3
2, 1
0,
?
-
PI * 1
P3 +1 ) P.
2,
71
*r
):
1.)
2
1) (
· 1 ) [ - p3 · 1 3
-------------------------------------------
[
*t*L1UATION NO.
ZR3 = - P.' 3. 1. i,
- --
1
3, 2,
1,
1,
0,
R2 ·
-
0, 1, 0,
I ) P2(
I1
I
( - R1 ·
(
1
-a2
0 ) R3 R1
P3
P.
I
1, 1,
1, 0 ) P2 P
-- -,,,, - - - --
I ) P.
-
, 11
0,
3,
R3 ·
1,
0,
, .
1
2,
,I
1, 1,
0, 0)
2 2,I
,
:
1,
P2 El
C2 , 2.3.1,I)
2
1)
) P.( 3. 1,
2
1)
( -
RI
I
-P3
1,
0,
* 1 ) P.(
0,
2,
* (
1 ) El
P).(
2P ·
-----
:
------
1I
:
P.
--------
3,
P1
1,
- ------
) :I
2,
,
- 32
I ) P.[
3,
2,
(
(
-----------------------------------I---------------------------""
IP3
4,
1
1,
2,
1) 1,
1
,
P33 .
I
P.C 3.,
- --
---
.
1,
0.
O ) P1
R3
P.(
8, 2,
1,
P.
~-ll-
-B
1I )
- P3
1 ·
t)
-I F2
1,
1,
1,
1)
1.
1,
-
0.,,
3 ',
P.( 2,
P,
3,
2,
3.
, 1%
---
---
1,
0,
12)
C
· - P1
1.
C -P
1i
2.
0
1 }
1...
....
33 E1I3 · ( : -·
2,
2,
1.
2, . 1, 1.
1
1.
0
12
t - P1
P2
---
---
---
3. 1,
---
---
1 ) 32
1, 1,
-
,1 .*
.,1 I ) P.
-
---
-
3,
,0 ) RI 833
---
---
-C
- P2 * I
-
1 ) P..
- P3
1,
--
--
-
- .7 .
1)
-
......
1, 0
( - P
-..
.
- I...
-
-
................
P3·
3,2
- R2
1I
0. 0,
1 )
P.C 3, 2.
C
3
1,
0
3
- R1
3,
3,
1,
) P1 P3
I ) P. C 3,
-----------
- RI + 1 ) P.C 3, 2, 3,
- 23
P2 P1
P3 .
· 1 ) P.: 3,
0, 1
) PI
C I
3, 0,10,)
-+11
2,
I.
1. 1, 0 ) Pi
C -
1, a ) P2
22
C - a2
3.
I ) P. 13,
0, 0, 0 ) 33 · C
- P3 * 1 ) :
-
1R
3. 1
0,,
I ) P.-
C - 31
R .) 1
0 ) p1 83
2
- P3 . I ) P. C 3,
3
3,
3, 0.
1,
I'
,1 I
P2
C
- 82.
P2
..
.
1,
1,
0,
· e2 "1
P3
, I, 1,
0,
------------------------------
21 1, 0, 0
--- --------------
I3) P.( 4,
P1 32
3
-------------
2. 1,
I ) P.:,,
[· ---------------
…
- P3 . 1 ) P.(
4,
-
0. O)
- R2 · I ) E
GO
P.
(· - 82
e
'-
(-
1, 0
- R
1
32 · I ) P.C 2.
-
I ) ?-C 2,
P3 + I ) P.
-
--
1,
--
0, 0 3 P1
1,
1, 0)
------------------------------
P3· 1
C
I
2, 2, 0, 0, 0)
P.
0, 04 0, 1 -3,.2
_)
--- ---
P..3.
1 +I ) .
-
-------------------------------
F3
, 0,
I)
.....
I )
II
)P
---
+ 1)
3, 1,
79
0, 1, 0 ) + P. C 4,
1, 0 ) P1
· ?P.(
P
-I p1
2
0. 0 )
32
2, P
) P2P
I ) P.(3,
P1 P3 + (
P2
- a2
I )
P
I ) P2 P3
I ) P. C 2, 2,
.....
-
-
2 * 1 )
·1 )P.(
1
t1, 1)
I ) P2 · E
) R2 ,3
------------
.
11 * I )
-
1,
r-l
8, 1, 0, 0,0
- - - -
(
2,
*
78
1 ) * P.C 3,
I·
· :
(
- R2 + I ) P
I )
C -P3
1)
P1+
1)P.:
+
**eEQUATION NO.
E·.O = - P. 3, 2, 0, 1, I 3
-
-
0,
1,
*e
0, 0 ) :2 33 21 · P.
----------------------------------------------22
-----
-(
·1
- ----
P2 R3 RI
0)
76
P. C 3,
I ) P.:
--------
1,
- a{
a
1, 1,
· I ) P.C 2,
1. 0
2,-1, 0.
-
-
1)(-
**EQUArICN n0.
Z = - P.
3, 2,
P.(
P2 P3
- P3
2.
-
·
---------------------------------------------
-----------------
1-'
*,1) I, P,
P.I
) P.'4
1 ) P. C 2,
) + P.32,,,1
-
1, 0 ) P2 R3
- E2 + 1)
·
1)
)P3
-
,
+
·- P
1
0
1,, O,,1) I ) 221 R
- P1 + I 3 (
- --------------------------------
~--------------------'--------------------------------
---------------------------------------------
P1
I
I
P
*
c--
1)
0,
1.
-P
2, 2, 0,
~-----
----
*-*EQUATION HO.
ZERI = - P. : 3· 2, C,
3,
I.
I ) P2 ?3
-R3
P. : 3,
O,
77
- E2 * 1 ) :
+1)(
-P3
0
. 0,
.
)P.:
. 1 ) h2
3,
, 1,
1,
C -P2+
3.
- P,
P.
:
3
)
1. 1. 1. I .I
3,
2,
75
P.(
1)
-------------
*** EQUATION
ZERO =
2,
, ) F.1
-31
- P2 ·
3
3P.:
, 1,
*I )
3P.
------------------
I )C
P.
0 3
0,
-
- - --
2C
0
I )
-.
- - - - - - - - - -
-P
R2 i3
P
0, 0,
3
*e
*-P.
,
£2 -
1 )
***EQUATION NO.
ZER3 = - P.
3, 1,1,
I )
- P2
0.
I 3)
e
P
, O0)
1)
1 · 1 ) C - R2 · I ) [
-
P.
C-
10,
74
3,
C
a,
1,1
-
) ?2 + C
1, 0,
O
(
f~
- - - - - - - - --
,1,
?.
I ) P2
·1 ) P.
I ) P3 ·
+1)
73
·
* ?P. ( 4, 1, 1,
I
· 1)
***EQUATION NO.
ZP O = - P. [ 3, 1,
:3 ·
~
3,L 2, ----0, 0,
I )
-------
1 ) P.'
4
.10.
- P.
p3
,,
- P3
***EQUATION
ZEDO =
- P1
-- "----"------------------ - - -- - - - - --
) PI R2
I )P
-R.).l
P.
3,2,),
1,
) P233 3
- 2+
1)
1+ I ) P. ( 3, 2, 0, 0, 0.)
3+
-+P3 +1)
7.43,
.......................................................................................................................
P'1 *
- .2 · 1 ) : - P3 * 1 ) P. : 3, 2, 1, 0, 1 ) P1 · C - P3 · 1 ):
- RI + I ) P.: 3, 2, 0, 1.
----
· C
al
- 'i
10
I
1 3·- ?o
P31.
1 )
R, . I-3
-- - -
1,0,
1C
1,0,
,
(
]
I ) P.( 3,
it
) P.
-
1, 0, 0 ) P I
- R2
+I
- E: · 1)
1 + f
1,
P2 P1 · (
)9
(
72
7·
1 ) * P.
3,
- - - - - - - - - - -
I)
7"*
( - B2 ·1
0,
P2 * 13.
3. 1 , 0,
* I ) P.(
1,1
)1
', 1 )
,
I
**#EQUATIc
ZERO = - P.
,
-R3t1)?P.C2,810OE+
3 *
I ,-P. ! 2,
41
·
***i.Q:UATIOP
::3.
1,
:
- ,2
I, 1,
-P3
·**ZUAT ICG l2).
.
Z -O = - P.: 3, O1,
) P.
+ 1 ) F.( 2,
69
n,
- R2 * ¶ ) C - P3
- P,2 ·
- P2
-P
·.1)
PI + I
- P3 · I ) P.( 3,
· 1 )
+
(
0 ) P1 + (- -P21)(
I
1,
I
- P.: 3,
.3 + (
1,
S,
l',-72,l)1)
+)* T - ?2
ZERO =
1,
0, I)
2
-
-
R1
3
- - - - - - - - - - - - - - - - - - - - - - - ------- - - -------------------------- P2
I )
- P3 * I ) P.[
, 2, 1 1, I ) P1
R2
.......................................................-
- P3
,
1 ) a2
, ,
3 ·
* C -
------------------
I 3 P.:
-------------------
P1 R22
1
1)I
"""""
1
C - )R1
, 2, 0, 0
0 ) 2I2 83
""
-
31
----
P.4.
2
1.
0.
00
C - P2
"'~~'"
""
""
1 ) P.
, 2, 0
0
----
----
· I
-----
…
I
----
----
+
1
)
---
-924
'EQUAT1iN
- - ['.(
P. ( 2,
2,
0,
1,
C
3)
+I
P3 · '1 - 71
+1
81
+ P.( 2,
NO.
3, 2, 1, 0, 0)
AEPO
1
*
1 -
2,
2 *1
0,
1, 1 )
(
-
:- ii2 * I ) P. (2, 2, 1, U, I ) P3
)
) :-821
)
- '3
+ 1)
P.'
2,2,
C -
P2P3 H1
2, + 1P3) P.(
1, O 0
P.
I2
1)
*
2,11
, 0,
4
)
2,
2, 0, O0 1
P3 R1
3.11)
(* -
*( - PI1 · I ) ( - R3 * I I
- 83
:
12,
2, ,1,
1)
4
I ) P2
1.(2, 2, le, 1, 0 ) P 2 · ( - PI
)
44
82
* E2UATDI3 N2.
ZEiO = - P. ,3, 2, 1, 0, 1 ) + P.
2,
P. ' 2,
+ I )
U3
3.
0,
i,
· '- ) · I
1 ) ;2 21 +
1
-
+ 1 ) : -21
)
-22
£2
1
)
444
P.
3, 0, 1, 3 ) 222 83
2.
3,
- P 3 + 1 ) P. : 2,
1, '),1, )
3,
1,
1
1 ) P.
PI
-
2,
A3 -+
(3 -
2 * 1}
3,
0,
0,
Pl
13
1
P.C 2,
E1 ·
:
3,
- P3
)
0, 0, 0)
83 a1 +
- p1 · I ) P.(
(
.I
2 , 3,
2.
3.
1,
(
- 93 *1
1,
1,
0 ) P2
1 ) ?2 +
PI
,) I
4
444
83
** FQUATIO0I 80.
ZEFO = - P.
3, 2, 1, 1, 0 ) + P.: 3, 1, 0, O, 1 3
2 p3 a1 ·
- P2
...... 7----------------------------------------------------------------------------------..................
P.
3
1, ,,, ), I )
£2E1 '
,- PI · 1 ) P.'i 3, 1 1,
1 )
:2 P3 + I
P3 + ' - 1 +
1
) : - P3 · 1 ) P.: 3, 1, 1,
, 0 ) R82
-----;---------------------------------------,**FQUATION NO.
84
*
zLRo = - P. 13, 2, 1, 1, 1 )* P.
3, 2, O0, 0.
) R2 83 al
(
- ,2
P.: 3,
2,
0., I
33
2, 0, 1, 1 )
* C - P2
2
1):
R1 +:
- P2 + 1)
1
-
+I )
1I
13
-3 P
--------------------,-----------------,-----,,-,
artEQUATION NO.
ZERO = - P. C 3, 3, 0,
1
- p3 + 1 ) P.: 3,
P.
3. 2,
1,
1,
) P.
3,
2. 1,
3,
1,
'1)
)
3 + C - P
'1 ) P. C 3.
- P2
1,
·I ) [
P. C 3,
+ 1)
2,
1 ) R1 +
+1 )
0,
1,
.1 I
0, 1,
-
- P3 +
*1
1 ) P3 21
- P1
0)
3,
( - i3
1I
-..............
1,
!, !,I )
2,
1,
*
8 381
3
1)
) P.C 3,
P-:
3,
P.C
( - P3
1
0. 0)
2, 1, 0. 1 ) 82 +
i2
- P2
1,1
a
85
+,P.C
3,
p2 P1 P3 · C - P2
P
1, 1
· 1 ) P. C 3,
0. 1 ) P1 P3 * C - 81 + 1
3. 1,
P.C 3, 3, 0, 1, 1 3 22 P3 4+
- 82 · 1 ) ' - i1
1 ) P.C 3, 3, 0, 0, I ) P3 · ( - 83 + 1 ) P.C 3, 3, 1, 1, 0 ) P2
-----------,------------------------------------------------- _,,
_,_,-,,,,,;,,
:--------,---,-----------,,,,,,,,,,,
P1i
- E2 + I)
: - 3 · 1 ) P.
3, 3, 1, 0, .0 ) P1i
- 3
1 )
- R1*
1 +e. C 3, 3, 0, 1,.0 ) P2
82
+ 1 ) C -.
3
1
) :
-1
+1
) P. : 3.
3, 0. 0, O)
**$EQUATION NO.
8644
ZERO=
- P.C 3, 3,
', I, )
P. ( 3. , 1, 1
--------------------------------P. { 3.
, O, 1, 0 ) 22 83 * C - 82 · 1 ) C -
) P2 P183
C - 2 · 1 ) P.C 3, 4, 1, 0, 0 ) P1 3 · C - R1
1
------------------------· 1 ) P.: 3. 4, 0, 0, 0 ) £3 * ( - P3 + 1 ) P.C 3, 4, 1. 1. I 3 P2
~------
P1 + (
- ;3 1 ·
) :
- B1 + I ) P.:
**EQOUATIO NO
0.
ZE8C
- P.C 3, 3. 3
1,
)
87
P.
P.
4, 2. 1, 1. 1 ) P P3 +C
- ·3
...............................
82
( - P2 + 1 )
3
I
P.:
*-*EQUATIO
=
=E -P.C
Z
P.
82
4. 3,
(
1,
NO
NO.
3, 3.),1,
2,
- P2
3, 0,
4,
4,. 2,
O, 1,1
1, ',
) i2
**
1 ) P1 82 P3 + C - R
I )
- P3
1,
1,
1 ) P.
, 3,
1,
4. 3,
89
P.C 2,
03
1. 3 ) P2 81
R
+ 1 ) P.C t 4, 2,
O, 0, I ) 82 P3
- P2
1 )
I ) P.: , 2, 1, 0, 0
P1 R2 · ( - e3
1 ) ( - 81
I ) P. ( 4. 2, o, o0.,
0. 0
..
........
.......................................--.....
........................-4. 2, 1, 1, 0 ) P1
0. 0)
1, 0 } P1 a3 *. C - P3 + 1 ) P.: 4,
**EQUATION NO.
ZERO
- - P. C 3, 3,
P.C
89
P.(
1)
3,
1
I
-
3,
1
1,
,
)
1
1.
C -1l
P182832
3,
1.
0,
41
) P.C 4,3,
0.
,
0,
8283
'
13) C .- 81 * 1
1 ) Pi 82 + C - P3
P.[
q,
(
-
3.
3, 0.
I)).
P2
1)
I
.
*$.
1 ) P2 ?3
I1 +
- E3 + 1 ) P.- 2,
3,
2 + 1 ) P. ( 2,
O, O, 0 ) 81
3,
0,
C - 83
0. 1 ) 23 81
R
- P1 · I ) P.C 2,
3, 1, 1.
1 )
1)
PZ
P3
- P1
I )
- 2
I ) P.: 2, 3, 1, 0 1 ) P3
: - P1
1 ) : - 83
1 ) P. C.2, 3, 1, 1,
0 ) P2
- P
-- - - - - -- - - - - - -- - - - -- - - - - -- -- ----------- - - -- - - - ---- - --- --------1)I
( - R2
1
3 + I ) P. ( 2, 3, 1, 0. 0
......................................................
**EQOUATION NO.
90
*
ZEEO = - P.:3
3
3,
1, 0., 1 )
P.
2, 4, 0.,
)2
381 ·
- E2 1)
P.C 2R,
. 0, 0 ) R3 1. I1J
l ·
- 3
P. ( 2,
83
----
4,
0,
1,
1
P2 21 {
(
- P1+
1 ) P.( 2.
4, 1,
1,
0 ) P2 83
C - P1 t 1 ) :
- 82
4
1 ) P.
2, 4.
1,
0.
0)
-
3. +.
C - PI1
) ( -P3
1 ) P.C 2, 4, 1, 1, I ) P2
-------------------'^"-------------------4**E2UAXION 10.
9144
ZER3 - - P. ( 3, 3, 1, 1, 0
+ P. C 3. 2. 0, 0. 1
82 23 81
P. ( 3,
2
, 0, 9,
0)
.2
{1 +
- P2
) (
P3
( - P2
1 ) ( - P1 .( I ) P. 3, 2. 1,
- ------------------------------------------------I)
-(
1·)-3
I ) P.( 3, 2,
- - - - - --------------------------*4**QUATICN 80NO.
92
ZE8: = - P. C 3, 3, 1, 1, 1 ) · P. C 3, 3, 0,
------------------------------------------------------------------P. ( 3, 3, O, 0, 1 ) P2 R1 · ( - V2 · 1 ) (
---------------------------- -- - - -- - 3
(-P2
*1
) (P 1
) p.
3, 3. 1,
· I ) C - P1 + 1 } I
***EQUATION
1.hO = - P.(
-r------
P. ( 3,
0NO.
3, 4, O,
--------------
4,
1,
1,
c. 1, 0 ) P2 :
----.-------------
- P3 +.1 ) P.! 3,3,
93
0, 0) ·
P.C 3,
-------------
0 ) .22
1 ·
(
4,
1,
-83
3
- 22
) P. ( 3,
1, 1 ) P3 · :
0. 1,
- P1
I )
4
1
P.
3,
0 )
1
( -P1
- 83
----
2,
', . 1 ) P3 i t
1 )P.
3,
1
P.
3, 2, 1,
------------------------
C
2, 1,
0. 0 ) 32
)
0, 1) 82
2
-
1. 1, 0)
------444
2 R3 81
C· - P2 · 1 ) P. C 3, 3. 3, 1.
,
----,----- P3 · 1 ) P.( 3, 3, 0, 1, 1 ) 1 4 C - P1 4 1
-- - - -- - ------- - - -- ----- -- --- -1, 0 ) 3
(
-P1
1P3
I ) P.( 3,
0, 0 !
1,
1,
1
**
1 ) P2 P1
· 1 3 C - F3
3 ) 83 814 C * 1
I3
-) P.: 3. 3. 1, 0. 0 3.. 3.
--- - - - --- -- --- -3, 1, 0. I ) 2
P2
1
C - R1
P3
-------------------
- 22
2,
· 1 ) P.: 3,
-------------
I ) P.
3,
4.
1,
-
0, 0 ) P1*
- 2
1 )
- 83
1)
( -1
I ) P. ( 3, 4, 0, 0,
0 )
--- ----------..---..............
. ..
- ...........
4.
0,
1,
I ) P2 P3 4
C - 33 · 1I
------------------------------------
-
3
4
I
-
81
1
P.(
3,
4.
-9344
9l4
4QU ATIOS 140.
'
. -=
-P.(
:-
3,
P.( 4, 3,
1,
1,
·2
P2
4
'
4
1,
4, 0,
P.(
0)
i'3 · (
1 ) P1
3
1 ) C - ?
* I ) P.
·**Q414T4ICU NO.
1) P.-
P
PI(-,3.
I 32
9~:
- P2 · 1){
)
.)
1,+ 1, 3 C
3,3
·. 1
'2 1)
-P
P. 3,1 , - ,
1
NO.
0 4.
3,
E**FQU.TION
ZERC=
- P.(
, O,
3
.
)C,
3,
4
9d9* 3
+P-
-**EQUATIOI NO.
ZERO
- P. ( 3,,
4
) P3
,· t,
31
.
1
3,
1I
F3
+) ·1·
=
I
1
-1·I
.CO 4
----------------------------***3QUATION£
***E
3=
NO£1.
P.R N0.
4,,). 1.
ZERO
= U-- TIC
ZERO
P.C
4,
,
·I 1
{1)
- - --
.C 3,
-)
IC ' P.
I) · ?.-1420.
-i.
3,+
{
I UATI0#
) Pi
NO.
·C.,O
**EQ
1,
3
C
)C
P
, 0,
) P2· 1
1
-3
+
P.(3
3,1
- ?2
1
1.1
I
3
·
1
)
) ZP.( ) -2 2
P,
I , 1+
.( 3, 4,
3,,, 1,
1, 5
O,
- 0 ) (a]·
.P1 · .I
. [
3,
P...........
[ 3,
0 (-2P1
I )
C,( - P2
) 12
P2
,4
) -1, 0.
-p , [
1 - 3, · ,
)
·I ) P.(
1
1,
- · .3
0 ) RZ
IPI.1
-·
[4) 1)PC
+
#. 0,
3, 0.,
) P
O0,
.
P1
)
~..
P3
-.
~
~
~
~
· I ).[
.3
-
I9.3
3 P.
·
·
. - 3,), 4,
C
C…----------------------…--, 2 )*
~
~
P3 P1
· I
PI
+
) [ R3 +
~
~
~
) ,3PI.
33 4,110·
-O,1
…-
-
--
, 1 .I
3
I 0P(.
0)
#, O,
)(~_B
(O, - 0
-·
13
……--------------------------*
0,-
.
O ) a2.191
O,
I
P
1)
·
I
-1
P
…-------------------------…------------------------------------…
0 ) R
I ) P.! Pa
:
--1,,.3 0
2,
2.S.
)R13) PI# P.I P. 3,
3· 0,
4,,..1
C,
0,
1
R.
+ P.
· 1P3
I)
C )--
01
) -P.[ +
pIR 3,
1
·2 1
) {P-[t
I ) P3
,P. C1, 3, ,1
· 1
C - 3P
, ) P3 · 1
1,
( ·C, P
-I+)PaI
~
~
~
9.42
-I
1,
~2 - · {
) [ 3,,-------------------------1, .-)IIaP.)
--104
**
I, PI 13·
P.I --------------------------------3, , 2,
,1, 1·
1P 4
3) P2
P2
0, 0,, 01 )
4 P.(
1,
1,
) · 103
~
RI
')
IPi+ 1)P.,
I P.-,0I
P.-l)
-
2*.I
I I } 2 [,
P I-----------
1,
·
1
.
}2# P.( 3,, 0,
.......
1
1, -"""'"""
0,....
I,
,
·- ·
)1 P.2
1
* (. -
P.(
3,
22
,1
C
P21..00
P1)
PRI33,
· I1) ) P.(
C
(1 I-12
1 3
1 )0.
1
P2
RI P.(
-* 1 ------ P.)- - ---------------3.1.
0,
1.I ) 1,, 0.
31,
I ))1, P3I
I(
-----
---I
0 ) P3I1
·
----- · --- --
RI .
-P212.2
--- ----
1.
i )
2p
1 [) .
--I
…-
----
l5*
1,
P..
0,, 1
3, 2,, 0,
0,-02)
-P. :1)
2
)
{ills*
- I P1·
·
, 4O,
2, 1,) P2
I
31
-P R2 3·
-I.2*l)P.1
- ·2
-
P2 9 1
3,
3,- 3,,
) C1
3
- P3I·
I ) P. [ 3,1,P
1.
-------------------------------------------------------10
* 3I
P.'
, 1, 1., I
1I) P2
t3
0O,
0, 0
1,
--
***EQUATION NO.
- P.C. 4, 1,
ZE.O
F.(G
0.. 0, 0
) R 2 Et3
) (
3
.1)
I)
,3 - I,,1,
-, . --) ,2-- 1,)----3
-- --,RI
Pl2 - -l I -P - [1 C
,O·C-)n )- .-2 -. - - -.- )
-31
-P C
- C-- -P. - .1,,1- O·
C- --E-RI
) P0
C.
O,
:1- · -I #,
) ?.
I· *
(
--
-
3.
…----…------P.C3.44.¶.0,0)32.:.............1)...C.5.........
-------------------------------------------------------------------------------------------------------------------------------0, ,0 0+)?
- I
R2
+ I I-3
- -2 + 1 ) P.C
3,
C,~ -llPC41001,pC
P
-g =1 -
0,
1
, 0)
5,
1PI
I ) FI
4. 3.
3,
p183 R
P.
O,
·.
Pi0, + II)0
.. 2·1
30.
, 1, 20,
T1IR
-1
4,
i1
I ) -1- P.(? · 1)3
(0 )
-I3)
P.II·.1.0
I -PI+ P.t,
· 1 ) P.
4,* +
1, 1- 1,
1 PI
I(
I 1 · 0 3, P
---------------------------------------------------------!
I.
1, 1,
0
a3 · [ ,,·I.
ZERO
2,
?I + ,
O, 1,
C -33.)
1-)
C…· - ---RI3
**EQUATION NO. ,0,,
1,
03
C3 ·- P., 0 4,0,1)p
)-ZER=
-------------*4+*P
3
C - P2
- )
· P1
S2
,
----------------------------4. 1, 0, 0, 1 )
3130 ·
P24
I
1,1, )
?I.I
) P3 1,·
2
3
------------------------------------------------------------NO.
I
1· 1, . ) 1.30I- . · P2
3, 5
0,
0,
4,
-
C
1, 0
C 1
1P..,
)
3·I)P'
1,
- P.'
(
4
4,
· I1 ) P.(
I ) P.{. 2, 34, 0,
-
--------------------------------------------------.
.
.
.
.~3.La.1.1,13p3.:
-P1411
-- 1**ELOATIo1
I--------------
1
1 ) R12 P'3
991
.
O,
1, .
-------------------------
80RO =
F1JC
,
3,
3 · IC.) (3, -,2 P3
P. O,2,, 1, , 0 ),, 1a1,3 · )[ P2
- E3 4 1 3,) S.
- P2
4
.-0,
I. I )
.I·I I) ) P.(
P.0 a,0
:
)3,1 .( 3,
,O.0,
1
3. 1?
,
1 ),
.l23,
- .I ) P
· 1,,
·P3 I ) 1) :2
.......................................................................................................................
..........................
1·-
1
P12333
-)
R
0.0
.:
1)
--------.
,I1P./25,
·
) P.
R I P,
3
1 )32
3,
I
-- R3
4,
*
3,
)
1.
32 · (
P1
:
,)' 21
1 ) 32 P3 1
1, , I'3 )
P3 ·V3 I · ) [ . - -E1,I·
·3,
2
P-P2) 3,
---------------4 ;.
· I p3
) P................
,')
1 3, -+
0, 0 3
1,
* 1 ) P.
-
22
I)P1.(
)P. - 413,
I ) P. C1* -2, P1,
- 1,1
C[I)) ) (3,{,, -434I)P.t3,0,1.o)i81t
1·-1
97
1 ) + P.
3
.
0 ) P.1
-
4I
96
P.(
1,
1
25
)
I, 31O,
1,
0- )
0,+
1,
2- ?I
**,Q'UATICNNO.
ZEfROC= - P.( 3, 4,
P.C
[
3,
,
1,
1,
2 P
PP
3 :
4ee
9.C-
4*thQUATICN NO.
41, 1,
Z'LE0
-P.{3.
.C)
3,
P(
P
2.
1, ) P.( 4,
P2 1, )+.II1)
-1, '4,
1
C
- [24, 4 1,
- P.( 3.
ZERO
) = PI
t ),, R3 4P,+ I,(
1I )0 l p.
P3
4,
.95
NO.l
4,4,1
· 1 ) P.(
.
P3
-
0. 1 ) P1
3· 1,
4.
-
3
-....................
:-
1.0
1,
0
P2
·
31
(
- 32
-P3+1)
1,,
I )142P. 1 3· 1,2· 1,I1)
- R I + P.C3
I
P 2 PI33
I
.2,...- P
I. I 2 C
…..........---'
3
0I.1 )
10,
3I 1, ·
----
·
1,,1.
---..........-
2,)
· I ) P.(
O,Ip2P 3
'
2 3,,
P. O·
1,
I
P.
)I
3, 2.
+
( {2I
P.{
P32-PiCI
-)
1
0,
- Pt·
0,,
0,
1)
(
( -
RIl
3,
9.2
P..{12,
-P.C.2+1)
·
1)P.(3
3
.
.................................
.....
-944
4'2~J4~I
C1
40.
107
*
"4'~3p~4
137*(
ZERO
= 12-- ,?.( N'E0J~.TIL{.O.
2,
1, ?I1, 1 1,II.;, 1 )P.(+
p .(4.4
--- -- 71
------+ 1 ,-------,----------,--E2 + (
- 73
Pi
· 1 )
, 1, -,
1, O,0 1P2
) +3 1 R3
·P. R,1 21.,P, . 1, ,+2
,
1. i3
' RZ
2
0,
-,
2130I ) I - - -1 - -P04. 1. 4.- 1- ,,-0,) 0
- ,1,
- - I, - O,
- - I -) i- - 5,,
- -0, - 2- - P.[
I,, --0, - I -} -P[2- - ( -- -[3- - ·IP L' S.
5
I,
I,
O, I { R2-
P.I
- PI · 1 ) C
· :1 - :12
E**iU:,ThION
ZEP..=
1,
- P.[ 4,
1
2,
r,
0,
2,
Ci
-·
0,
1 ) P.
.:
4., 2,
pi 1,
-1 2
4**,QuA'Jx'IC
44.
ZZRO. - - P.: 4, 2,
:
-1
P3
1
0
4
I
4,
2,
- 23 ·
!
[ 3,
2,2 0,,
P3C·
1,
1
0
-P2C
1
) 72
.1 +
-
P.1 )-
1
- t~2
-'
2
3
***EOA1'TiON'{ NO.
ZEO = .
.2,
P.:
I, 1
1,
:3 -
i
***EQTU.TIGN
ZERO =
- P.
P.[R
-
·
,I
) '.2 Pt3
1}
P.
N3.
14 , 2
q,
)
1
3
2,
3,
,
- 1.[
4,
3.
,3,I)
,-----O,---- 1, - I--) P2
----P.3, -3,
3 --------------'
1,
I, I
?3
+ 1 ·( I - PI + +I
)
3
1,
1,
| 4
I
) ?. :
P3 ·
m,
-------
***EQU.TION NO.
ZERO
= - P. . 4,
3,
1,
P.:3
P2 R 1
3,
2-.,,
1,)
', I
i
&ZEo =
-
.
P. ( a.
3,
I
- P 3+
) :.
P.
1,
[
4,
I1)
3,
03
1,
· P3
·(
, 1,
1, I3
4,
1,
1,
1)
i
P.(
-
0 ) P3
13
(
- 142 {
C- 3 3
1I
.( 4,
4
·I
P..
3,
C - P3
) P3 R1
·
0, 0,
- PI
) P.(
3,
2,
1.
1,
)
3,
0 )
1I
*
1 )
2,
1, 1,
3, 1, 0,
1)
P1 R1 · ( - P3
_-.......................
1 } P.I 3,
3.1
3,
P3
i ) P.:
-----
.0.
- '1·
?2 · (
?2
- P1
, 1 )
·1)
2 +I -
0.1,,0)
I ) P.[ 4,1
- Pl3
2 1
1,
1.
I3
P3 +
I.)
- P
R2
+ 1 3 P.
3-
1P
4,
3
3.
2,
1 )
0.
? I.C
---
32 31 + {
, 2.
2, 1,1
2,1
*
1I
- L3
P
3
..
P+ .
P1
-
- 4 141~
*
1. 1.1
33
- PP3
1I 1 . C.
P.
P1
P2* 1· 3r
- P2,*2.1
---------------------------
*
P2
1 P3
3 CP.C
5,- *
2, 11 3 0 · )
P2
it2-C
-P2
. C,#,
.,1,3,
3 ) P2 PI · 83
·I *
3,
- a2
,
C
- P2
·
I
) P.
4s,
tl,
I,
,3 ) 1,, ---,a
-l+(-P
-I3
1.I ) t,, 1,PI
· ""
-------------
+
,
,
P
0 )
3·1, 4,
2 R3
·
1
1
O,,0
3,
1 , 3,
['2 + (
R1
+ [
2
·I
- R2
O!
1
------------
10)
P.
I31
1 3
C-
I i
O ) Pi G{
·[ -
O,
"
23 -"?2I
-
. -"
·3.
-R
"
· "I
I1)P. (5,
· I
::
,?5
[·
2
J -·
-
R3
) -P
-e )
3
3
0
1 !-+
I )
[ . 3,
0·
P3
O , .
.PI:3
1P+R
I
2,
10)2+ -
-I3) P.[Ca4
·I
3.1.,,0
P.1s,
2+21
3,
4.
----
R2 + (
-
I
4,
12
·
3,
I
, I ''~"
t.1)P...........
P.( 4I iS
0 ) R3
,
IP.
·I
I
t) 3.
?2
3,
- P3
4.
1,0.
3
7
1+
1,
+
I1}
1,
0 )
I....
I)
O}
- PI
.I(.
I-
I )
-1 ......
---...
-
22 R2I
27,,
- a3
----- 1,
03, ) 1,?2·[-
0O, OI_}_R2_Rl
7PI ( I ) P.[C#,
I1)
R1
- )
3, P I,, .[ 3,1,
PI
- · 3,I
P.([
4
1P-2
·--------------------------------I ) :
- PI + 1)
21
.-
) P.:
2'3
P.
------
--------------------------------------------- R3
+1)
P. ( 4.2,
3, 0, . 3
[
) a2 a3 R~I
- P3+
-
- O ------O,, "0
3, -3,
P:
, +
- I ) [
P.
0 ) R2 R3 +
,31I·
---------
I **·
) ?P2 P3 RI
· +
,
3 4, q
1 , 0,1I)
I ) P.14
1 ) I,
1? +
P2 +
0, 0,
O, 0
[
.!
P.
R3
3,
2,
- R3 · 1 I
4
- P2
1- *
P2 C I
3+
+R133 ) PI )
* ,_
"'
R2
I)
3,
.I '+ (
13,
E2 P3 P1 + 4
0
{"----------'-'--------------4, O .2,
1, O,---------------------------) E2+1,
119
1,
a )
- 3
5C 1,
----------------------------.3
-3) I P'3
) ·P..P.2_P.
[ 3,I 3. 3, }/1,
.3,1, O, I0.)
7**
111
)
P.
+
P. ( 5,
0,
- P2 3 · 3I
+I
112 3
0
-----------------------------------""
" 1, "0, '-I1)
R2?23
+ P. I5,2,
P. [q,
1,
· I
3
1, 0,
^---------------~---------------------,3,
PI
0 ) 72 £3
O,
**
}0,
2,0
,,1,,1)·?.,,3
,',
- 42 + 1
P.(
(
2,,
- P1 +
P3" +: " - P2
1)
- a2
1 ) P.( 4, 2.
-
I
( -F2
---- :
P.
O,
3,
I ) P1 P33 +
I1 )
- a2 4 11
- P2 · 1 ) C
:
) P2
2 P
,4 2,, ----1,
4,
I I ) ['
3,
------------------------------------------3 ***EQUXTION
· --P1
·I -----------{
3
·1
) P.(
3, 4, 1,
1, I
SO.
11
ip*
ZERO
=
-?P.
C4,
3,
1, 1,3)+P.[t
4,
2, C, 0,
I
1-+ I )(------------. q,3+ 2,
1.
P. ( 3(,*EeQU&TION
NO. 1,
- R2
**
+ P.(
)
1 ) P.(
1, 0,
3,
I ) R,2 P3 + :
3 ?P.
R2
P.( 3 4,
P 2} EI 2 +
/ ·I :
:P.3+1_}
(
I }
)
O 1 P2
O
4.
-,,,,,
1, ) 116
+· P.
9,
,
1, 0,
, 2,
+
*4
5, 2',
--,--------,-----
----------,-- S3.-,-r,,,
***EQI{~TIO{
ZLR3
= t , 3.
· (
+ 3,
1,
I )
0 } I3
1, 9,
3,
i , RI
P3 +
I 1, 115
+. [I ?.[
0,
1, 1 0 ) P2 '1
O)1
114
,·
- a2 + 1 ) ?.(
, 24, 02, O0, 0 )
P.
3,,,
I,
1, O,
P.:3+
,3 I ---------------------------------) ,3)P
P '--------'-------'-----------------------------'
4
4, .1+:-R
: n, 0 ) R3 , ):-8
,[
----R1
R,
1,
,. 0,
113
* P.C
,0 , 3, ·
1 1***,Q/ATZON
-0 ) Ii
1)4 +NO.
P.::2 4
PI£,
1,
P.:
- i 3 ·
· 3,
112
P.
,2 .,3 + P.
NO.
, 3,
***EQCUTION
ZERO = 3 - P.
(
0, 1 3 P2 P3PI1
0,
2,,, 1,1
3,
I}~
P.C: 5,
1
3 ·
1144
,*4
P.', 3, 3, 1 1,
)P2.1·
?2.
............................................................................
I-12C
} ·
, 0-
, , 1.
3
1 P3*. ,.Z ?3 ·, 23 1.0. -1,3 P2)
P.
, 14
13 4,1 .4,
- ----- --- ------------------4, 2, 1, 1,
)
I )
3: P.,
2
4, 3,
0 ) P2 P12
P3 R3
1,
.[3,
~1).3
.
P P2 1
R1 · 1 )
- P1
3
+ 1 ) 1.
·
1
4,1 3,
I---------------------
ZEEO=
, 1)+
'-
· 1)
I
-
3,
PP.2
·
1. 1 )
: 1 }:
1,
- 13 · 1 ) P.
+1
t - I2
2 - F.
1,
1,
, 3, 1,
i 2
***ZQU£4TZON
ND. 2, 1,0
ZERO=
-P. ',4,
.......----------0.3 ·
PI, 4, I
1,
**
i P.
P3
3,
- ?1 0 I )
+1 3
4 1
11,9
0,
0,
? 0,
1 ) 1, .:.,
1
P.(
O, O, 1 ) [3 · (
I, C 1) -P 2
:1
1P. 3
.............................................................................
·**.'Ui-;!3'T
.
O;,
N.
11
ZER.O = - P.: 4, 2, 1, 1.,0)
4 1.( 3,
P.
) P.:
1,V8
8to
NO.
1I
-7 Pi..
I..-.
Pil
3 · ( -Pi
·....
( - PI· 1I)
( -!'{
....................................................................................................................-_.
-95-
13
,]-,QIUATli N 4*
=
.1. 4,
ZErO
,
0.
120
+'
0
0,
$**EQUATION NJ.
ZEO
- P.! '4 .t.
0I
.1
---------------------------------------------------------- P.
-ER:
( '- }:3 1'
'4,
,
1' P.(
3,
4,,, O, 04.. :)
' - P1
0 )2 +
, 1,
**SEQiJA'TIOi NO.
ZE.P: = - P. ', ,
1, .),
P. ( 4, 3,
1
3,
,1.
!
)
1.1,0
-2
1, 1 ) + :
1,
.
1 )
3
I ) P3 + (
) P. ( 3,
4.
'2 P3 R1 *+
2 P3
- (
+I
I0
-
I ·
1.
-P P1
,4. 3.
) P. {,4
1,
-
- a3 .1
{
*1)
2
3, 4.
) P.(
0 )
0.
1
- PI *
3,
3 ) P2P 1 +
1)
I ) P.:
S.
3
4. 3. 1,
3,,
- FR3
3 P.(
( -,, 3 · 1
) P.:
) P2 P3+
- P2 + I ) '
1.1,
1. 0., 0
'.
P.
4.
PR3 1, 0)
4, 1,
1
1. 1
3,
0.
- P2
0, ,0 ) R2
3 )
3,
C - P1 * 1 )
2 RI
1.
) P3 * I
1, 1
(I
. )
-
3+
*4
- P2 + 1 ) P.. 4, 4,
E.3.
- P1 4
(
-+ 3
+I
"
) P
1 ) ( - P3 + I
?2
82
P2 3 .
)
(-+ -
C
R3 21
P1 . 1 ) P.( 3,
C
1)(
p
5,
P.(, 4
-P3+. 1)
3 1
1+)
+ ( - 3.
.R
1+) P.( 4, 5S 0.1.
4
O, 1. I ) P3 E1 + C - R3 * 1 ) P. C 4. 4. 0. 0., :
0
0
4
P.
1
, 0 )
, 4, 1
-----------------------------------
4
- R3 + I ) P.(
4,
22
0
1 *·
) R2
4, 1,
- P2 + I ) P.I 5,
P3 +
4
1
) P.( I
-3
+
--
) a2
, 1,+
-P1 + I
2 +I )
-
0
a,
0-
.
- P
1 )
-, P2,
,4
5, ) 0. 1,
P. 5, 4. 1, 0. 0)
I
-
:
F.3
.
*
1 ) P.C 4, 5, 1, 1. 0)
0-I ) F
. 1,
P1+
P
I + P. ( 3.
2 R3
0
P. [
125
)
(
1, 0 )
1,
-
.I2
I ) P.
) +
-
1)
-P3
+ I ) P. 1,.
+ I1 )
-
: 21)
1
, 1 ) P.C 4, 1, 0. 0. 0 ) E3 R1
( - R2 +
-I ) P.( ' 1.
- 32
1 ------------------------------------+ ( - P1 + 1 ) C-------, 1 ) Pt.
--1, .,
4,
C - P1
, 0 ) P3
1I
+I )
3
-
3 · I ) (
I
-
1.
I) [.
0)
1.
127
P.
1
' - P3
4
3. 4., 1I, 1. 1
1)0
0
-----------
3. 1)
I
126
5,
------------------------------- P3
2 + 1)
5.
1
-PPI+1)
)
E**4QUATION 30.
ZERO
1 ) P2 P3 , (
it2 · I ) ( -
#*
I )
) +:-
- El e. I}----------------141)-----------------
3
O, 1 )
O,
P4I-
2
3.
+ 5,
- F3 + 1 ) P.
P3
1
*
O, 1 1
0,
124
***EQUATION NO..
I
ZE.O = - P. 4 5.0, 01,
-
3
--
. )} _2
1.1.
I,,
- 22
-+
3NO.
**PQ[IATI(N
- P. ( 4, 5,
ZE:O
I, O . 1,
---------
- SI ' 1 ) P.(C4.5, 5
:
1.I + 1 ) P.'
-
3,
-)
3
P
*e
I )1
2
,,! ,
1,
)PI
)
0, I , 1 ) -.1 +
.4,
4.
3,
1,
t,,
1 )
123
0 ) I P.'
:1,
-**EQUATION 43.
4, ,4. 1,
ZE.Ro
- P.
p..
2
-
4
P.(
- R1 · 1.)
_R34+1)P.C4U1O
- --
.
1,
+.
.2 P3 + P.( 5,
1 j
+ 1 ) P. I,4, 3.
3, ,
3
-1
1 ) ( -
, 1,
- P.3 + 1 )
} (
121
I ) + P.I. 4, 5,
(
1 ) P2 P1 '3
I )
- -- --
122
1, . ). 0 ) + P.:
NO..
**EQUAT:O4
1,
.. 4), 1,
---
P. (t ,It2.
I,4..1.0J?2i'14
- - -- -------- ,---------------------------'
).
)
1 ) P. [
'4..- 3
1,
P3 + I ) P.C 4,
1)
-P3+
P.#,4.
1,
1,
1.
+[
,
11
1 ) P2 * C
1,00I,)
1)
0. 0.
I
C
-
0.
) R3 · ' (
O, 0
1.
1
- P1i
-R2+
4.
) P.(
1
11
I
-.
+ I ) P.(
- R2-----
32 + I ) P.( 4.
0. 1.
0.
1
.
-R2+1)
P.{5, 0.1.,0,1)
)
------""
------------------------------------"--------"
" ------5--------,1, 1,
0, '---""
"
"
- - - I'"
"
'- ***EQUIATO
----------------"--N3.
- a3
1 I P.C 4, 1. 0. 0, 0
ZEPO
- P.
5, 1. 1, 1. 0 ) 128: - E2 * I ) P. C*4. 1. 0, 0, 1 ) P3 31 · C - R2 + 1 ) .
4..
129
***EPQUTIC, NO.
P.
4, 2. 0.,O. 1
P3
1)
+
}C
2
0,
0,
0
)
R3
2
I
)
P.:
4.2,
1..,
3,
1
)
:
=
-P.
5,.
ZEO
'------------I----""
""
---------------------------------- -"--'~"~------------ ) 17-------P.
1 ) P3 * C - R2 * I
) P. --4.
ti,1.
1. 1
) P2 P3 + C - P1 + I ) C - R2 + I ) P.C 4. 1, 1, .,
4+ [
- P1"'- + I ---------------C5,1
) - --------3+
,10,.I
-P1+1)-
-------
------------:
-------E21)
.1,1.0.,0) ------
-R3+.1)------------P.(4,
------------*
-
-
----------R2$1)P.
)
- 1 )
--- I -) - P.- - .3" 4 "
--------------,-- 8.- 0-- ) --R3 -4 C -- -R2 *
C - -P1--i
I1 -) - --- 12-- 4 - I -) -P. -4. -2.-- 12
'. ---------------2.- ----1,1.,
1
)' P2
El1 +
- P1 +
----- -R I1
- +
-------------- 12 + 1 )
1 )2 P3 1*- C
P.C 4, 2. 1. 0. -------- P1i
1 )------------C - R2 + I )---------------1. 1, - 1)
P2 P3 +
P. '4,
2,- ----- P1 + I 1 ---------------'~
"
"
--------------------------------)
-P3 " 1)"
C
-P P 1)
C
1 )
- P3
1)P.C
4. 2, 1. 1, 1) P2
[5. 2, 1, , 0 } [
- P1
?3 +1, 0,I)
1- )(- ( 4, 2, 1, 0, 0 } * C ---- ***EQUATION
---------------10. 2, 1, 3~
EIO
. = - P.: 5,
***EQUATICN NO.
5. 2. 1, .
ZEBO = - P.
C5, 2,
--4, - -7
.(
2, - 1,- 0,
)F1+ (
5,
·
4. 3,
P.( 4. 3,
-P11I)
1. 0, I 1 }
[
( 5,
3,
1, 0,
I1
-
- P1
1 ) P.C
2,
**4
1 ) P.: 4, 3.
R2
P23*
(
0,
1 ) P3
0,
0,
0 ) S3 al + C - 32
-P14
P
1
+
.
0,
-iI2*
1)
- R2
1)
-
-''3 3
-
I ) P.. S.
0
"--'-----"-----------------
-----
I )
C
- F
+I
[ }4,.
P C3
3,
--------
1, 1,
...
1 3
C
- 33 · I
}C - P3
+I
4
P.( 4. 3.1.0. 0)
C -R21)
1)
1-3 13
I)
P.
4.
I ) P. C 4.
3* C
2, 23.,
3,
3.
-12+1)
I
0,
0. 1
P
C -P3*1)1I
I )
P3 E1
- R2 +1 ) P.( **4
4, 3, 0, 0, 1)
1., I ) P2 P3 · ( - P1 ' I ) ( - B2 4
.IC --------
4
-21P
3.'1,
-P 2
'
C-P1l
0 ) P2
4, 2,1,1,
C -P3+I)P.(4,3,1,1,1)P2C
- F2 + I ) C.
?3 + f.
r-
- E2
1,1,0)
-----------------'------ -------132
***EQU1ATIO
&EEO
.. - P.( NO.
5, 3, 1. 0, 0)
* C
) El + ( - .1 + I ) P. ( 4 , 3. 1,
.C #, 3, 1,
-------------------
-R3 + 1 ) P.C
"-'
- - 5,- 1),
I
( -- -'
P3 - - I ) - P.:
2, -1,- - -----------------------C - P.3 + 1 ) P. C 5, 2, 1. 0, 0 )
----------------------------4
-PI1*)
-?3-(
3,.1.,0,
PI + I
-"R2 4- 1)
P2 4 1 )
- -- 130
- ) * .
131
I1 )
:
P.
P.
P
P
C24 1)
4.i 3.
I ) P.
O ) P2 + (
- ,33):
)P.
5, 3. 1. 0. 0
.....
--T -.
..
-------...........---....
- P14 I
C -1 i3
1.
0.
t
C
I1) P.o I4. 3,0.
0. 0.
P
IF.
I ) P3 + t - 12 · I
- R2 + 1 ) (
-
3 4I
P
-96***Q(!.TION NO.
P.: 5, J,
Fz.C
- P1
1,
+ 1 ) P- ' 1t,
, 5 ) E3 *
- 21
aI'2ITION. 110.
i,a, 1,
inO = - P.-
133
) *
o,
+ 1 ) ?P.(
-
9, 0o4,
1 ) P. t 4,
- P3
134
p.(
, 5,1,
4,
**
1 )
1,
- P2 * I ) 1P.
1,
I ) P2
P2P3 RI
4, 4,
1,
, 1, 1 ) ?2 P3 +
1, 1, 1 ) P2 P3 + P.1 5, 4
, 4,
* 1 ) P..
O. O )
4, 4, ,
) 2.
f1 +
1
-2 F
I- )
- P:
+
.
.
.
.
.
-....................................................
,
135
4t*E 'I.YTICi NO.
1
(
1
LPFC = - P. t 5, t, 1, 1, 1 ) * P.( 4, 5, o, 1, 3 ) R3
- P
4,
- PI + 1
1,
h3
.
5,
1,
.
1)
P.
.
1
5,
* ('
,4, ) E
O
4,
1,
1,
0 ) P2
t, 1, 3, 0 )
----.-------1,
0 ) R3 * P. [ S.5,
1,
4,
) P.( 5,
*1
1 ) P2
{( - R3 * I ) P.{ 4,
- 2 +1 )
.
.
1 ) P. ( 4,
1,
) P( .4,
--
2 * 1 )
+(
, 0 ) 1 3 * ( - h2
*(
1
I )12
1,
,
4,
P.(
P .2
) P. ( 5,
- ?3
*
1
-3
O ) a3
- Pl + 1 ) :
1, 0 ) P2 E3 *
1,1,
1)
*t
4 , 4,
I,
0 )
S3
136
-*cEQUATICN NO.
'5, 5, 1, 1, ')
'ZfiC = 1 ) P.
'3
+ ..----------------..---.--
-
5, 5, 1, 1, 0 )
_--_____..
+
I
P.
**
4, 5
0, 1, 0
l(
)
PI1
-3
-
1 ) P.( 4,
*I+
5
1,,
, 0 )
APPENDIX B
E(s,U) Expressions which are Non-Zero and Not of Internal Form
(See Tables 3.1- 3.3 for transient states and boundary states with
expressions.)
internal E(-)
Edge States
E(l,n2,1,1,lU
)
n
= X1 X2 2 Yl(l - r2 + P2 Y2 )/P2
2,...,N2-1
n2
n
E(l,n2
r 1p2
1-r 3 + P3Y3
n2-1n
2rlP
~l-r
3
)
P3Y2
+
= ;11X2Y2(1-r3+P3Y3)/P3,
n
E(nl,l,l,l,l,U) = X1 X2Y2 (1 n2 1=2
1
1
1
)
r2p3
=
1
P3Y3)/P3
+
3 3p3 -
3
X2Y Xnl-n=
2
n~~=
(nl,0,0,0,1,U)
iAn1 ,O,l,O,l,U)
3r
1-r3
--
3
2
'
l-r3-P
[ lr2
1-rr2P
p+ Y2 1
N2-1
1
32
2,..., N1-2
n
l
, 1
(1 -I
r
212
n- -1X2Y1 Y
1 rnl3
1
= 2,...,
1
1
N2-1
r2-P
-
n
n
5(nl,l,0,1,1,U
.
2
2
3
2
2
),
(2-Pl
(1
2n1
21 n
,X -1
(O,n,O,,O,U)
2'0'1'0'U) =
n=2
r2 + P 2 Y 2 )Y 3 /P 2,
Y(1 -
= X1X2
,1,1,TU)
U -r
p 2 )(-r 3 -P3 )j
nl =- 2,...,Nl-1
-l xl j
l1
n1 = 2,..., Ni-1
-97-
,
-98-
N -1 n
= X1
,U)
,1,1,0
(N -l,n
(1-rl + PlY 1 )Y2 /P
X2
= 2,...N2-1
2n
1,
N -1 n2
=
(N-,n2,1,1,1,U)
E(Nln
2
1,0,0,U)
2
I
-
r 1
2
= 1,...,N2-2
-r+3P Y3
(l
Plr 2
2 33
3
x 2 Ja3
xl
=
(n,N2-1,,1,1,U
)
1
= X1
S(n 1 ,N-1,1,1,1,U)
2
-p )(1p
(1-r 2 +
2
X1
N2 1
p
1
2
3
n n-
(nl,N 2,0,1,,U)
(1-P 2 )]
(1-r-P
2
n2
-1
n
N
n 2 = 2,...,N2-1
+ PY1)Y2Y3/Pl
(-r
2
1
2YY
/P
3
3
3
N2Yl(1-r2 + PY
X2
3
2 2)Y /P
= 1,...,N -2
1
n
2
2,
n1 = 1,...,N1-2
n= N2-1
1 - r2
(n 010,1U)
r3
i-rl
X
, -12l2Y
=x
2-,lO1,1,U)2n
nl N -1
(OlO
X
U(nlN ,1,0,U)
1 2Pp
=
1
1
+
_
2p
3
X1
X
1
n(1-r2 p
P2)(l
3,
= 1,...,N
1
Y1
2
2
2 r3
2
L1-r
1
+ pY
1 1-2-)j
(1-r-p)
n1
=
3
1,.,
Corner States:
1 -r
2
rlP 3
(0,0,0,1,1,U)
(r
(1
+ r 3 - rlr 3
1 -r
(0,1,o0,1,o0,U)
-
-12
XX2Y1Y2
121
YP3rl)XYX2Y1Y
2
N1-2
-99-
3rr
3
r
r(
r2r+)2
r--P-
r
,rI + r 3
(0,1,0,,1,U)
r
P 3 rl
=
xlx2
(
2 +rp
(1,0,0,0,1,U)= 3(rl
-
-
r= +r
=r
E(1,1,O,1,1,U)
r lP
Y2
= X
P
1
Y
+
+ (1 - p 3 )
r 2 )j
- r3 )(1 -(1r1-)
-
r
3 11 +2)
3
3
U(1,Q01,1,,U) =
(1
p
(rr-
x y1 y
12
3
(1 -2 r 3 + P3Y3)/P3
[(1 - r H(1- - r 3 ) + (1 - r+
12
=
O1,1,1,1,1,0U)
P 2Y
2
)PY3
N1-1
x1
x
P1 P 3
(Ni-,1,1,1,1,U)
(1 - r1 + P 1Y
1 )Y 2
(1 - r 3 + P 3Y 3)
Pr1 P-1X2Y
2 3 2
S(N1,O,1,O,1,U)
E(NlO,1,O~lu)
i(N-,N2
0,1,0,U)
=
= x
=
2
P1r2P
3 [
N-1 1X2
X1
-1-rl)(1-r3
) (1-rl-Pl)(1-P2 ) (1-r3-P3
(l1-2)(l-r3
P1
N2-1
N-1
X2
(N1-1,N2',1,1,0,U) = X1
1 (1-r
1
+ PlY1)Y33
)
(+rl)P2r 3
(r+.r3 - rlr3 ) (1-r2 )
2r
(1-r)p
(l-rl)P1P1 1 2
-100N1-1 N2-1
4(N1 -1, N 2 -1,0,1,1,U) = X 1
X2
(l-rl
N1-] N2-1
= X1
X2
(l-rl
i(N1,N2,1,1,0,U)
12
2
(1-r)
+ PlY1)Y3 (1-rl)P2
(r +r
+ PlY1)Y3
1
(N1,N2-1,,1,1,U
)
E(Ni,N2
x
= X
X2
)(1
2)
(1-r3
11323
N-1 N
1
rr 3
(1-rl)PPr
3
(r
(1-r +
rPr3)
3
1Y )Y
11 3
( -r )
r-
(-r)pp2r3
,-1,Q,0,,U)
=
N -1 N -1
x
1Pl(1-Pl-r r)) )(-r+
2
3
(1-r1
1
rY
3
1 (r
2 2 + r 3 -r 2r 3 )
APPENDIX C
U
k[ s]- Expressions
Limiting
The limiting solutions of the parametric equations (4.1) and
x(k)
(k)
(k)
(k)
(k)
(k)
(X1
zero
,
2
k(S),
, Y
2
(4.2),
), as well as the corresponding non-
for k = 1,...,12 are listed below.
For each case, all other
Ck(s)
tend to zero as U + Uk).
Each limiting case is indicated by the italic numerals and arrows
in Figure 4.1.
Case 1
X
9
O0
X
1 +
X
0 }
x2/xl
+ Q2
+ 0
Y1 +00
(1)
2
2
y
2)
Z2- (
p2
--r3
l-r3
+y(l)
33
P3
where
+
(l-r3)
Q2 ( 1
2
)P l
)(1-r3 -p3)
and
(l-r 3 ) [l-r 3 (l-p1 ) (l-P 2 )( -r3 -3 )]
Q2 =
(1-pl)(1-r3 -p3 )(1-r
)(1-r
-(pl)(1-r -P )(1-r -P)
Limits:
1-r
( 0
,0,0,1,1)
=
2
(rl + r3 - rl
1 r3
rlP
3
1-r
(0,1,0,1,0)
=
1r
(2
-.0J--
rl3)y2
-102rl
=
E (0 1,0,1,1)
'1-r
+3
1
=
r13
(1)
2 2 3 (1)
5
3PlP2Y2
+r
= p
(1,0,0,0,1)
1
3 (rP1
r
2
1
2)
r + r3-rP
13, I,133
~1(1,0,1,1,1)
= Y2p
51(0,2,0,1,1)
=1Case
-+(1) r
1 ~l(0,2,0,1,1)
=2 r~p
32
[ (1-r3P3)
(-
[(1
3(l)
Case 2
XX
22
Y1
0
0
+
y(2)
1
+
Y
'((2)
2
Y
where
2
+
0
1
= _
1
1-p
+
2
2
p
(1)
(1)
3
)
)
(-p)(-r
2 -p2
-103-
+
Z2
1'-p 1
2
(1-r 1 -pl ) (1-r3 )
and
(l-P 2 )Z2
-
(1-r2 -P2)
+2+ 2
Z2 (Z
-
(1-r2))
(0,0,0,1,1)
=
Q2
Limits:
1-rl
2
r1P
(rl
+
r-
+
r
r
3
rlP3)
rlr3
2
3
(2)
1-r(01010)
1
~(0,1,0,1,1)
=
2
(2) (2)
-n
(2)
31
1
13
21
rP3
(110,0,o0o)
=
3
3P2 _(2)_
+ 1 - p3
2 (1,0,1,1,1) =
r1+ r3 -r 1 r 3 -r 1 P3
(
r
1
2 (1,1,0,0,0) = 1
(l-r3)
(1,1,1,1,0)
2
=
y (2)y(2)
1
2
(2)
(
(2) (2)
1
12
(2)
(1-r3 -P3 ) (1-rr)
2
-104-
(!'r2)(!r3)
2(1'1'11'1)
(2)
1
=P2P3
) (
(1
-P
52(0,2,0,1,0)
=
-
52(2,0,0,0,1)
=
(!-r3
.1
r 2 p 3 (lrl
~~2
~2
1
~2
)'
1
~
r 1 P2
(l-r3
r2P 3 (1-r
52(2, 0,1,0,1) =
1-r-P2
P )
(2)
(2
2
p1
(2) (2)
) (2)
1
2
Case 3
X1 +0
X2 + X23)
y
> y2( 3 )
y 2 + 2(3)
= 1/Q
3
l-r2
3 P2 (1r
3 )
+ (1-r2)
3
Q3 (1-Pl) (1-r2-P2)
and
3
where~r
Limits
3(31
p
rl3
( 2(
(1-r
3
)
2
P3)
2(3)Y2(3)
-105-
rl+r
r
3 (0'1',0,1,1) =
- 3rr3
X23
p
r = p1
3
r2
2 2
3
rlp
3 (r1 + r2 -
r lr 2 )
1 3
r 1 P3
1 3
1
=
i(1,01,1,1)
3)(3)
y2
(3)
X(3)
r p p X
3 (1,0,0,0,1) =
2
3
3
(3) (3)
2
2
(3) (3)
E3(1,1,1,1,0) = X2 Y2
(3)
3(1,1,1,1,1) = p 2 p
(1-r2)(l-r
Case 4
X1+
Y2
0
X+ X ()
2
(4)
3
1/Q
+
Z3 -
3
(1-r
3)
P3
where
+
(1-p 1)
Z3
3
(1-rl-P1 ) (1-r2 )
and
(l-p 3 ) Z3
3
+ +
Z. (Z3 -
(1-r3 -P3 )
(1-r3))
3
+ p3Y3
) + P 2P 3 Y 2
Y3
-106-
Limits
-p
P3(rl
4 (1,1,0,0,0) = X2
E4(1,1,0,0,1)
~~4
p)
3(1-r
4(1,0,0,0,1)
=
+
(1--rl)
r 2 - rr 2 )
2
(4)
2
4)
X (4)Y)(4)
2
3
(1r +)((4)
r 3 +P 3 Y3
2
(4)
(4)
2
1
X (
4p2P3
For 2 < n < N2-2,
(4)
n-i
1r)
[
2 2
[1-r 3 + p 3Y 3
(0,n,0,1,0) =
~~4
5 4 (0,n,0,1,1)
r1 p 2
(4)
2
3
1
r
(,n,0,0,1)
X2(
P2
(l-r2
=
P2
_2
v
X
(4)
2
(4)
Y
(l-p (1-r-p)
1
2
2
2____
(lr2)(4)
Y3 )
-r1
2
4 (0,n,1,1,0)
=
)
(4)
_
X=(
E4 (l,n,0,0,0)
4
n-1
(4)
(4)
=
(4)
1
(4)
(4) (4)
2
1
3
-
(1rP
2
-107-
f
4 (0,N2 0,1,'0)
(
2
_____=
p2
3_
_2
i r3
2
= 2 (r1+ 'P(-rlr3)
N -1
( 4)
4 (1,N2'1-1,0') 0) =
N2 -1
(1-r2
~4 (l1'T
N2 -1'0'1l'
1 l
-r
E4 (1,N2 -1,1'10) =
N 2 -1
(
x (4)
2
P2
2
(4
(4)
N -1
N2-1 (4) (4)
Y
(4)
(1-r )
(1,N2-1,1,1,1)
(4)
) 2X(4) P2
(-)
=
2
P2
N2-1
y:(4)
Z(4)
2
)YX~
(1,N2,O,1,0)
K
3
2
2
P2r
3 (1-rl + PY(4)
(1-r
1,1,0) =l(4)(4
(1'N2'
)
4
Case 5
Xy
+
X2+
015(5)
0
Pl
Y1
(5)
y3
3y
-
(l-r3 )
p
P3
-1
24)
P2r3 (1-r 1
4
+
2.
(4)
)
iYl
Xl
(4)
3
-108-
where
+
Z
Qi1 (1-P
(1-r 3 )
2 ) (1-r3-P3)
and
3)
(1-r
=Q1-p
2
3 -r
(1-p
3
[1-r
3
-(1-p 2 ) (1-p
1-r
1)
)L(1-r
3
(1-r
3-P 3
)]
(1-Pp
2 )
3
-r
1)
)-(1
3 )
(1-P 1 -r
1
]
Limits:
(1-rl)
r3
1(rl+
2
5 (0,0,0,1,1) =
.(5)
- rlr
3 -r
P 3 )X 1
(5)
r 1 P3
f5(0,1,0,1,0)
5
1-r1
=
rl
(5) (5)
X
Y
(5) (5)
(r + r3 -r r3
lr3) X(5)(5)
3
1
1
1
r1 P 3
(0,1,0,1,1) =
(5)
(5)
Y1
r3plP2X1
Y
5 (1,0,0,0,1) =rp
;5(1,'
= (r1 + r3 -rl3
1,1,1) ........
~5
5
1
1r
(5) (5)
1 ' ''
1
(
- r1 P
3
(5) (5)
P
(5)
3
N 1 -1
X (5)
S5(Nl,1,1,0,0)
=
r-
51pir 2
(l-rl)(1-r3
(1-r
-L3 - P 3
-P )(1-P
2
-109N 1 -1
5(Nl,1,,1)
~5
= 11(
1)
=0
(l-r
) (
1-r3-r
3 -p 3
p1 r2
1'~
3rl)
)
N1 -1
r
(5)
(N
(1-r
1l-r)
=
,0,1,0,1)
3)
-
(-rlP
1
) (-P
Case 6
(6)
= Q1
X1 + X(
X2
°
v + v( 6 )
1
1
zl
+
Y3
r2
1-P
=
y(6)
2
2
(1-rl)
-
2
0
where
(- P 2 )
+
1 (1-r
2
-p
2
) (1-r
3
)
and
-
(1-p 1 )Z1
=
Q1
+
(1-r 1 -p 1 )
+
zz1 (Z
(Z1 - (1-r
-
1
)
Limits:
(1-rl
~6(0 0,0,111)
(rl+
-2
r 1P
3
( 11
(0
6
1
0
1
0 )
=
r3 -r
-. (6)
(6)
1
1
(6)
2
1
(6)
r 3 - rlp 3 )X 1
(6)
1(6)2
(6)
2
) (1-r
3
-P
-110-
6
rlP3
1
1
Y2
3(6).
(6)
66 (1,0 ,0 , 0,1) =
3 1
P 3 (rl
+
r2 -
r lr 2
)
(6 ) (6)
2
r1
+
2
1 - P3 - (1-r3 -p3 ) (1-r
1-r1
~6(1,0,1,1,1) :
(r + 6r
r3
P3)
- r()3
r 1p
3
3
rlP 3
1-r 2 )
(6)
X (6 )y (6)y (6)
1
1
2
E6(1,1,0,0 ,0 ) = X6( )
(6 (1 l
(l-r)
-
X3(6)
(6)
P3
6(1,1,1,1,0) - X
~6 ~ '=
(6) (6)(6)
1
Y2
(1 (1-r3
(6r2)
P2P3
(1-P1
n-i
(6)
For 2 < n < N1 -2
6(n,'0,0,0,1)
)
. (6) (6)
1
1
(1-r2-P 2
()
)
(6)
r
(6) (
=
2P3
x(6)
56 (n,0,1,0,1) =1
1 y (6)
y(6) (6)
1 2
6r2 p3
3
l
1-r. +pY(6)
)
(1(
P
3 r3-
3-3
(6) -P
(1-r3
1
2)
3
j
6(n 1,,0,0,0) = X(6)
6 (n,'l'0'') = -
X3(6)
p3
(n
6(n',1,1,0,0)
.3
1
(6)
Y2
(6
(6)1,1,oo)
(6)
= X1
y 1(6)
(l-r3)
p3
(6)n (6) (6)
1
1
2
N1-1
6 (N1 -1,1,,O,O0)
-
6 (N
=
1,1,1,0,0)
I
1
(
N -1
(6
~N
(l~3
(l-p)N-p
1)
'6
,P3
0)+
(6) -1
(6)
:
(6
-1
)X1
Y.
-r
(-r
)
N -1
· (6) ~(6)
6 (Ni,0,1,
'I1 P r p
2
-
Case 7
X1 +X(7 = Q1
x2
P2
7
Y1
)
Y2 + Y ( 7 )
2
Y2
3
c
1
(1-r 1 )
1-r 2
3
[
(1
i
1 -p1
)
-P
-112-
where
+
1
(l-r 2)
Q1 (1-P 3 ) (1-r2 -P2 )
and
(1-r 2 ) [1-r 2 - (1-p
p33 )l-p
(l-p3 ) (1-r 2 -P2 ) [(1-rl) (1-r2 ) -
1)
(1-r 2 -p2 )]
(1-p 3 ) (1-r 2 -P2 )(1-rlP
1
)]
Limits:
(1 -r)
(N
N
7 (N.l,N2-1,0,1,1)
=
N 1
2)
-(l7r
71 2
(1-r')
.
(1-rl
+
PlY1
1
1
N -1
___(7)
p1
-1
)X1
1
2
7(N -1,N2 -1,1,1,1) =
(7)
(7)
(7
y (7 )
X1
1
p1
(1r(7-r
-1,N
~7S71(N
- ,0,1,0)
22
=
-1
(1-r)
1)
(1-r
)P2r3
(1-r1 + plY
)X1
(7) (N7 N -1
%7(N-1,N 2'1,1,0) =
2
r1 P3
= (
l + p
2
7(N1 ,N 2 -1,1,1)
P'r3 (r
(8)
X1
=1
x+
2
00
Q1
N -1
X
+
r3 -r2
r 3)(1-r
(rl+ r 3rr3
-Pr3
1
..
2
)
(1-r )
.
(1-r )p 1P2r3
Case 8
13)
11
Y(7 )
(N1N -X1,1,00)
=
2
rl
3 +
(1 )
1-r)plP2 r3+
(1-r2)(1-r
r
1
N -1
(7) (7) 1
(1-r1 + p1 Y1 )X
1
-113-
Y1
Y2
Z1
+
+
y(8)
1
(1-r
p1
0
(8)
3
3
r3
1-p
l 3
where
+
Z1
p
=
3
)
1 (1-r ) (1-r3 -p3
and
-pl
'
~:(z(1
)
- (1-rl
Z
1
1-1 )
Limits:
,0,
t8 (1,N 2 -1,
0 )
0
= X18)
1'-rx
8 (1,N2-1,0,1,1)
=
(8) (8)
Y3 )
X
2
82
1
(1-r2
(1,N -1,1,1,1)
8('2
=
1()
(8) (8) (8)
(l2
1
P2
-
(8)y(8)
x ()
P
8 (1,N2-1,1,1,0) =
3
3
1
For 2 < n < N1-2
S 8 (n,N
2
-1,0,0,0) = Xl
(l-r2)
E8 (n,N 2 -1,0,01,)
=
1(8)
2
(8)n
1
(8)
(8)
3
-114-
_
(8)
(8)
(l r2
(8 )
Y3
Y
X
(nN -1,1,)
1-P2-(1-P2-r2) (1-r
(0 NP0,1,0)
= 22
1)
3
(1-r3)
1 3
For 1 < n < N -2
()
(8)
;8(n,N
2
0,1,0)
l=
r3
(8)
(8)
(8)
1
1
3
8 (n,N2 '1,1, 0)
r
+
l-rl
2
1
P1
(1-r2 -P
13
1
(8)
1___2
2
3 2
1-r 1 +
(1)3
((8r2)P)
-1-r1
2
(8)
1( 1
(1-r -p
2 )2
x1
-
(1P 3)j
N1 -1
8 (N -1,N2-1,0,0,0)
= X1
(-
8
8 (N1 -1,N 2-1,0,1,1) =
+ p1 1
1
y
)3
N -1
x1(81
S (N -1,N -1,1,0,0)
8)(8)
X~
31 2
,-(N -1,N .0,1,)
8
1
+l r-r3
2Pl1-3 r 2)) (1-P
+ (8
(1-r
1 )p 1P213
NN~
(8)+X(8)
2
-1,1,1,0)
58(Nl-N
ll(
=
+
(1r32(1-r
pY
1-1
P
~1
(8)
X1
l(1-r
(8) .~ 1(8)
+-1
'(1-r
r+
1+
1
(8)
)(1-r 2 )
p1 Y1
-r
+
~-
(8)
(8)
(
3
(8)
3
8(N
(rl+r3
N 2-ll, 1,1) =1
r3
p
-
(l1r
)
12
(l-r2
rl
S(0,N
8
2-1
,0)
=
2
)
-r3
(2 ( -r + p y(8))X(8)
3
1
(8) (8)
(1-r3 + p3Y 3
(1-r2 8
rl P 2 (1-r 3 + p3Y3
X2)
(8) (8)
(8)
Case 9
X2 + X9)
2
=
/Q
11(9) =
(9)
Y3
1-r
P!
Z3
3
3
P3
where
(1-r 1)
+
=3
(1-P2 ) (1-r
l-P
1
)
and
(1-r) [1-r
3
Q3
=
(-P2)
(1p
(1-P 2 )(1-Pl-r
1 ) [(1-r 1 )(1-r
)
3)
(1-rl
P1 )
-(l-P 2
) (1-r-P) (1-r3 -P3 )]
y(8)
-116-
Limits:
(9)
(9(N1,o,1,01,)
23
=
I(1-r 1) (1-r
P
g (N-1,0,0,0,1)
r
9 p(9)
1 2 (9)
(r
2
-X
"9 · ' ' =-1lO~l
Y1
p 3r 2
Case 10
-00
X
(10)
X21
+
X210) = 1/Q3
2
(10)
~y2~~
Y3
-
P2 1
3 - (l-r
(10)
3
3)
p
P3
3
where
1l-P
+
Z3
2
(1-rl) (1-r
2
-P 2 )
and
(l-p
Q3
3
) Z
z+
+
-
(1-r
z 3 (Z3 - (1-r
3
-
(1-rl -pi) (1-P 2) (1-r3 -P
3
(lrl)P(l3
=
(Ni-1,0,1,0,1)
3)
3
))
-p 3 )
(1-rl-P
)
(1-rl)
1
(1-r -
-
(l-r
)p
3
1
p
1
3
p
(-P
2
)(1
) (1r3-3
-117-
Limits:
(10) (10)
·, , 1,
t0(N1,0,10
1
2
P)
2 3
=
l(-rl)(l-r3)-(-rl-Pl)(1-2)(
-
r3-P3
. (10)
1O(N
0
1 -l,l,O,O,i)
= X(10)y(10)
(1-r ) (1-r
0)(10) +
(1-r 1Hl-'3
+ p3Y
3 3
=- (10) (10)
X2
Y2
P P
For 1 < n < N -2
x(1 0 )
(10)
(-rX
l-r3+ p 3 Y 3
X
E1 0 (Nl,n,1,0,1)
=
(10)n. (10)
(10)
Y -y
2
2
P r2
32
[ (1-r )X
[
For 2 < n < N -2
-2
(10)n
E10(N-l,n,0,0,0) = X2
E
(N -1,n,0,0,1) =
10 1
(10)
(10)
2
3
X
(1-rl
~101 ,2,
p
Y
(10)n
(10)
2
(10)
3
(10)
-r3 + P3 Y3(10)
(1-r-P)(1-p2
1
-118-
N
(10)-1
10O(N
1 -1,N 2 -1,0,0,0)
= X2
,
(l-r
3
X
(110(N-1N2-1 11
2,p
N10
1'
3
X
2
P2
-lN2-1,0,1,1)=
1O(Nl
N -1
(10)
(10) 2
1
N2-1
(10) 2
X210)
2
+
.(1-rl)10)
10 N1,N
2
-1'
1,1, 1)
= P
2
(10)
2
(10)
l
(10)
(10)
N -1
(10)(
X2
+
Pi(r2 r
31 (
3 -r2
1 r3- p r)
10(21'
Sio (N 1 ,N-1,1,1-1)
2
3_-.
=
2- 31
1
(-r) 3
P
1
P2r
)
10
10
(
.
N21
-2,N
(N
2Np
1-
')
=
1r
-1
(10)
(10)
N2 -1
(10)
2-r (10)
3
(1-r
(N
Nl03,L3
10(N1-1,N2,,1,10)
(
r
(10)
3
2
2=x)
1P2
-1r,01
(1-r 3 -p )3 )
(r(+i (l-p
2
=-2
1
P3) (1-r2 -P2 )
=
2
)(1-r
3
N2 -1
)
21
(10)
2
2
3
(10)(10)
2 2
x(10) 2
21,0)
(10)
(10)
-119-
Case 11
+ 00
X
+
X2/X1
P1
1
P2
2
2
y
Q2
Y3+
0~
X
where
(l-r 1)
_
Z
Q2 (l-p3 ) (1-rl-P1 )
and
(1-r 1))[1-rl- (1-P
(1-p 3 ) ( 1-r 1 -p1 )
Q2
2)
(1-p
3 )
(1-rl-P 1) ]
1-r 1 ) (1-r )- (l-P 3 )(1-r1 -p1 ) (1-r2 -p 2 ) ]
[ (
Limits:
11
1
=
(N1 -2,N2'0,1,0)
rp
3 2
Case 12
1+~
Y1
Y22
+
1 +X2/Xl
Q2
3
X2 +
0
2(12)
(12) (1-r2)
Z22
2 P
2
) (1-P
[ (1-r 2
[
1-r-P
1
(1-r2 ) 2(-p
3
-120-
y
. (12)
-
3
3
1-P3
where
+
(1-P
)
3
Z
2 =(lr)1 (1-r3-P 3)
and
Q=
2
(l-P 2 )Z
22 -
(1-r 22-p 22)
++
Z2
(Z 2
(1-r
2
))
Limits:
:12 (N1
N2-1,N
-10,0,0) = 1
(1-r2
E12 (N-1,NN2-1,0,1,1) =
(12)
2
(3
(l'rl
12(N 1-1,N
(12)
P1
2-1,1,1,0)
2
(1-rl
12
(N
12
(12) (12)
(N1 -1,'N 2 -1,1,1,1) =
1
'N2-1,,0
0)
2
P (r
2
+r
3
3
- r2 r 3 )
(1-r
1-
+
(r + r 3 -r
) (1-rp) (1-r
(l-r2)
r3 -Pr3 (1-r 2)
12 P2
2
r32
~-,N,0120)
(
E12(N1-2,N
3
(1-P3)(1-r2--P9
r2'0,1,0)
p
r
32
r1
(12)
3
(12 )
3
3
Y(12)]
2
)
-121(1-r2 )(1-r3 )
12(N1i-iN2
1
2 0 1,0)
;12
12(N1-2,N
1, 1,10
)= =
2P
3
2
3
y (12) (12)
3
P l P2 r3
S12(N
'N2 1'''0 ) =
E12 (N'N
1
2 -2'1,0,0)
(rl + r3 - rr3 -Pr3)
2
Y
=
(N(1-r)y
(12)
2
(12)
P 1 r2 (1-r3 + P3
)
3
(1-r )y (12) y(12)
12 (N1 N-2,1,0,1)
=
2
(12)
P l r 2 (1-r
3
+ p 3Y
3
2)(1-r
((1-r
(12)
-122-
APPENDIX D
A Computer Program for Solving the Three-Stage Transfer Line
Problem, with Sample Output
The FORTRAN program written by S.C. Glassman and I.C. Schick to implement the solution described in this report, and a sample output, appear in
the following pages.
Numerical experience with this program running on the IBM/VM and
Honeywell/Multics systems is discussed in Pomerance [1979].
-123-
I;::rJT CII.''G3E OF'PTION: 1, 2, OR 3 (II
.1
IiNPUT NI AND N2 (I5 FORMAT)
*
5
5
INPUT R'S THEN Pa' (3F8.6 FORMAT)
.20
.20
.15
.10
.05
.·
05
N
R
5
5 0.2000 ' 0.2000 0.1500
:JS T ilS CORRECT? O-YES.
.0
FORMAT)
0.1000
P
0.0500
0.0500
IMACHINE
1 FAILIJURE F'PrDABILITY. o.100oOO
REFPAIR FROBABILITY
0.200000
EFFICIENCY IN ISOLATION 0.667
MEAN UP-TIME
MEAN DOWN-TIME
10.000
5.000
MACIHINE
2 FAILURE PROBOABILITY 0.050000
REPAIR PROBZABILITY
0.200000
EFFICIENCY IN ISOLATION 0.800
MEAN UP-TIME
MEAN DOWN-TIME
20.000
5.000
MACHINE
3 FAILURE PROBABILITY
0.050000
REPAIR PROBABILITY
0.150000
EFFICIENCY IN ISOLATION 0.750
MEAN UP-TIME
MEAN DOWN-TIME
20.000
6.667
LINE EFFICIENCY
0.559302
EXPECTED STORAGE LEVELS AND FRACTION OF MAXIMUM STORAGE
STORAGE 1
2.3175
0.4635
STORAGE 2
1.8667
0. 3733
TOTAL EXPECTED INVENTORY
4.1843
PROBABILITY
PROBABILITY
PROBABILITY
PROBABILITY
OF
OF
OF
OF
MACHINE
MACHINE
MACHINE
MACHINE
1
2
2
3
BLOCKED
BLOCKED
STARVED
STARVED
0.1610
0.1144
0.1900
0.2543
IT IS
F
THAT THERE WERE NEGATIVE PROBABILITIES
ORIGINAL SUM OF P WAS 0.632Q+01
DO YOU WANT FULL PRINT? O=NO. (I1 FORMAT)
.0
DO YOU WANT A PRINT OF C AND U? O=NO.
.0
DO YOU WANT TO COMPUTE P - AP (O=NO) ?
.1
THE SUM OF THE ABSOLUTE VALUES OF P - AP WAS 0.296314Q-20
THE MAXIMUM ELEMENT OF THE DIFFERENCE WAS 0.9789830-29
FIRST AND LAST 2 SINGULAR VALUES
0.1646190+07
0.1218250-02
DO YOU WANT TO GO AGAIN? O=YES. (I1 FORMAT)
.1
"Sample Printout"
0.2713930-29
-124-
2- ------ ----.XAIN:
C
TdLS
IS THEt
---
---
----
INT£iiACTIVE
tlREE-rA.;E TiANS? ER
- DiIViL.
------
.iAI00010
d.i323 3-
------------
FDR THE!
MAI0030
iAL33O)4
5 10050
.hI)jO6~
LINE PirOBLEM.
C
iP?ITTEN BY S.C. GLASSSN .1hD I.C. SCHIZK,
LA3C£RAT)EY FOR I!;FOc ATION At5) DEBC1SION SYSTEMS,*tI3':73
ZA$SSZiV:JSI.TS INSTitUTE 5P TFLiN3LCOGY.
PFrOi;RA.
C
¶AIOO080
m.AI".s J9
C
FOE Taf35lY, SZE:
EAIOv,11
S.B. G'AS:iWIN AND I.C. SCHICK, 'ANA.LYSIS GF TRANSFER LINES
CONSISZING OF Tdl;EE UhiitLIABLE IACHINES AND
,TDFINIrE SIC0AGE JUFFELS' , '0OMPLE.X ASTZELALS
HA';DLING A:ND ASSEIl3LY SYSTEMS, FI:.NL REPDE.T,
C
C
C
C
1X',
VOLdJ:
INFOhi,-ATION
C
ESL-FS-Ie34-9, LABOE.trOY¥ FOR
AND DECISION SYSTV.S FtOrMELY
MAIO0120
.AIOC130
kIu140
-AII3153
AII.'160
?,,AIOc170
aAI(~18J
MAIO0 190
AI0v20D
2----------------A
EtLEicONIC SYSTE$'S LABCE.ATSPY)
ZASSCtifUSiTrS INSTIi!UTE OF TECHNZLD)Y, 1979.
C
.
ThIS VERSI3N: 22 MiAY 1979
I00100
C
c
tAIG0I22D
COSO.N /i?ARalS/
F '3), P 3)
t;.tAX :2),
AX
4)
IAI
BC23'i
.,AIO%.240
I NTi;E
IN EG ;i
3o:DSTAT(150) ,JCGL '150, 11), INL£X W'15a)
NFCDD, N?SIZE,IOPT, IGOG,N3)DD,'STATE.
N1AX ,IA,I,J,IIP,
IEr h IN DXVL. IPRINT, IUC, IAP
iLAL*16
FEAL*16
EIAL*16
REAL*16
REAL*4
FIINW, FMINW2, W (150),
P, P
D(150)
T(153, 11),
B :150, 153), KSI 3528),
'XJ13)
*
C
UFZERO TJ.NS OFF THE UNDERFLDI
C
-C
5 V1(150),
U :153,
DUIM MY, Pa03(3528),
5)
DJTPUT
.
.
.
CALL UFEEBO
lIP = O
NFODD = 15)
NPSIZE = 3528
C
10
C
C
OnTlINJE
WiI TE :6,930)
R E D 5,933) IOPT
.
.AIO00440
i AI 0'450
Al100460
1A13D470
MAI~'I483
~AIOC 490
MAILU5*0
3AI00510
IAIO520
IF IOPT = 1 THiN N:EW N, B AND P
IF lOPT = 2 T;EN JUST EVY I
IF ICPT = 3 THN JUST NEW B AND P.
3) GO TO 20
:TiT . L.
W.ITE 6 ,91 0)
F3AD(5, 320) NiAX :1) . NMAU'2)
IF
IF
ICtPT
C
2) CONTINUE
.EQ. 2)
GO TO 33
aAI0v250
AIA00260
lAIOJ27)
?1&AI 00280
SJ?P SA103290
5AI033C)
3AlOC0310
. AI I320
.'AIOG3-30
S AIC'34$
AO
I10u350
!AIO0360
AI00370
,qAI033'0
ikI100390
AIC OA400
MAIC#415
AIA0420
Alu:'43O
"AIO0530
A130540
A I0u550
-125WHRITE (6,920)
fiEAD(5,810) DUM(1),
DO 348 I-1,3
348 I (I}=EXT(DU (I))
READ(5, 810) DULi(1),
DO 349 1=1,3
349 P(I)=OE XT(DUH(I))
DUiI(2),
cUH(3)
DUN(2),
DUM(3)
C
30 CONTINULE
WRITE(6,990) NMAX(1),
WRITE (6,1000)
RiAD(5, 800) IGO
IF (IGO .NE.
0) GO TO
NODD =. 4 * (NMAX(1) +
NSTATE = 8 * (NMAX(1)
NBAX(2),
R{1),
R{2),
10
NMAX(2)) - 10
+ 1) * (NMAX(2)
R{3),
P(1), P(2),
+ 1)
P(3)
MAI00 560
HAI0C570
MAI00580
MAI00590
MAI00600
MAIO0610
HAI00620
MAI00630
MAI00640
AI00650
MAICC660
MAI00670
MAI00680
MAI00690
LAI00700
MAI00710
C
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
INIT(KSI, NSTATE, NODL, ODSTAT)
FORHU(U, NORD, NFODD)
FORMT(T, JCOL, ODSTAT, NODD, NFODD)
FOhftBX (B., U, KSI, T, JCOL, CDSTAT, NODD, NSTATE, NFODD)
EXMIN(NFODD, NODD, NCDD,B, W, IIP, DUMMY, IERh, RV1)
SCANW(W,B,NFODD,U,NOLD,INDXVL,INDEXW)
FOM PX (PROB,U,KSI,B(1,INDXVL),NODD, NSTATE, SUMP,NFOID)
I NFY (NSTATE,PROB, SUMP)
C
WRITE (6,9 30)
READ(5,800) IPRINT
IF (IPRINT .EQ. 0) GO TO 40
C
WRITE (6, 960)
WRITE (6,970)
(W (I) ,I=1,NODD)
-WRITE (6,980)
WRITE (6,970) (B(J,INDXVL) ,J=1,NCLV)
CALL PRINT(PROB, NSTATE)
40 CONTINUE
IF(IPRINT.NE.0) GO TO 50
WRITE (6, 101 0)
READ(5,800) IUC
IF (IUC .NE. 0) CALL UCPRT(B(1,INDXVL),
U,
NODD,
NFODD)
C.
50 CCNTI- UE
WRITE (6,1020)
READ (5,800) IAP
IF (IAP .NE. 0) CALL PNINAP(PROB,
CALL PRINT(KSI,NSTATE)
C
800
810
820
900
910
920
.
KSI, NSTATE)
-MA0
IF (IPRINT ,EQ. 0) WRITE(6,940) W (1),
W(NODD-1),
(NODD)
WRITE (6, 950J
READ (5,800) IGO
IF (IGO .EQ. 0) GO TO 10
STOP
FORHAT (I1
FCRhAT (3F 8. 6)
FORIAT (215)
FORiMAT(//.'
IIPUT CHANGE OPT1OY: 1. 2, OR 3 (I1 FORMAT)')
FOLIAT('
1NPUT N1 AND N2 (15 FCfiMAT)')
FORMAT(' INPUT R"S TIIEN P"S (3F8.6 FORMAT)')
MAI00720
MAI00730
fAI00740
MAI00750
MAI00760
!AI00770
MA100780
MAI00790
MHA I00800
MAIT0081G
MAI00820
AIO100830
HMAI00840
AI100850
MA100860
MAI00870
MA100880
MAI00890
MAI00900
!nAI00910
MAI00 9 20
HAI00930
MAI00940
MAI3095C
HI00960
HAIC0970
AI0 0980
MAI00990
IA101 000
1010
HAI01 020
M AI0 1030
MAI01340
MAI01050
MAI01 060
MAI01070
MAI01080
MAI01090
MA101100
MAIC1110
MA101120
-126) Yn)U WANiT FULL PFL;;T? 0=NO.
'11 FOFi.i'AT} ')
F3Ith;Ar r'
YI. ST A'!D LAST 2 SIN3:JLAi VALUES ',3j15.6)
POR. i2:
F3RMAXT
DC.YOU WAN'T TO 30 A;%i1I? L=YES. '11 FI).IIAT) I
FOGZAT(/,' $SI>GULAh VALUES OF 3')
FORMAT' ' 7Q15.7})
FORiAT(/,' C(J) ViCZTOr)
F 3 T.iAT"
N
*
/, 21 5, 38.4,5X,3Fe .4)
')
FE . C?
1003) FChA'l
'
'
S TItS :i t=YES.
1910 FC -0MV1('iOD Yi)U WAITT A PSINT 3F C Al;D U? O=Ng. ')
:A3=N)
?')
.
ANT TO COMPU3TE
- A?
1 320 ?r, ,MAr" DO YOMU
1END
930
940
953
960
970
90a
990
P'.
Pi
JAI01110
.AAI;)1120
AIO1133
IAI01 140
iaI$1153
MAID1160
AYI'1 173
Al 01180
H¶AI1 190
3A131200
AI0110
?.AIO 1220
-127-
ALEGC010
LC LSCAL rFUNCTICN AL(GAL(N, M.ACH)
ALE00020
C
ALE09030
.---------------------------..--.........--- ---------C
ALEOC040
:C
1T'iIS FUNCTIOIN DECIDES ;HET'rEP~ O$ NOT A GIVEN STATE
ALEOO050
( N (1) , N (2) , `AChi(1) , MACH (2) , MACH{(3)
C
ALEO00060
C
IS A FrCtI/.1SNT CR ThANSIENT STATE.
ALE03970
IiTAhSIENT
1S S-3ADY-STATE ?RCB5ABILIY IS ZEPO.
IF iT IS
C
ALECC380
C
ALE00090
C
'N;S:CiL' IS IiH STOFiAGE A.f-AY ?ADD-D I'IT3 FICTITICU3 S:OQEAGFS
ALECO100
FCFR:EP IS NON-E-:?TY AND THE L4TTEF NON-FrLL.
C
1 A.;D K+Il. iH
C-------------------------------------------------------------ALE0110
ALEO0120
C
CQ-O:,O:;/?UAAS/ . (3),
CCT.~C'.c
?(3),
1.'1IX(2),
A.LECj130
!.AXST(4}
ALEO0140
ALEOC 150
AL003160
ALEO0170
ALE00180
ALE0)190
ALEO0200
ALEOC210
ALE0C220
(4)
/ST(E S/ NSTCO
C
.
I:N7:E£
iN1 ,:*S
:N(2), ZACIt(3)
N' .(, N!;STCR,!A\ST,IND3
c
::AL*16
R,P
C
AL-kGAL = .TIFUE.
uC 11 IND3 = 2,3
IF
(;;STO£z(IND3)
IF
(.ACid(IND3)
(';STOr(lND3 -
01'iF
O)
ALE02 3.0
GO TC 102
ALE00240
1) .
O) GO TO
0Q.
%LE00250
11
AL500260
(I,;D3
.- Q. 2) GC TO 101
1 .AND. ,IACH(IND3 - 2)
(:STu- (I;D-3 - 1) .E.
IF
101
.NE.
.EQ. 9) GO TO 1
IF
IF
GO TO 1
.N-v . 1) GO TO 1025
ALE00290
ALE00300
(4ACH(IND3 - 1)
GC Tr 11
102 IF (NSTC, (IND3)
1) G0 TO 11
ALE03270
ALEOC280
.EQ.
1)
.EQ.
-LEO0310
1) GO TO 11
([ACli TND3) .EQ.
TO 103
GO TO 105
IAXST(IND3))
(NSTOCi(IND3) .NE.
(IACH(IND3 - 1) .ZQ. 0) GO TO I
MAXST({IND3 + 1)) GO TO 11
(NS$Tou.(IND3 + 1) .E.
3) GO TO 104
(IND3 .E2.
(NSTOF(IND3 + 1) .EQ. (.'AXS.(IND3 + 1) - 1) .&AD.
MACH(IND3 + 1) .EQ. 1) GO TO 11
*
1 4 IF (!"Cd(lhD3) .EQ. 1) GO TO 1
GG TO 11
IF
GC
1025 1'
IF
1'!3 1F
IF
IF
135 IF (NSTOF.(IND3) .RE. (IAXS (IND3) - 1))
iF (ACH(IND3 - 1) .EQ. C) GO TO 103
11 CCN=INUE
C
C 1F A STATE GETS TC HERE IT ISi ECURiFENT
C
ETLTFPN
1 CCNTINUE
C
C CTnil WISE T TS TPANSIENT
C
ALEGAL = .FALSS.
}ETI--J N
C
END OF
E***
END
LLEGJAL
GO TO 11
ALEO0320
ALE00330
ALEOC340O
ALEOC350
ALE00360
ALE00370
ALE00380
ALE00390
ALE00400
ALEOO410
ALE00420
ALE0O430
ALECO440
ALE00450
ALEO0460
ALE00470
A.LEO00480
ALE00490
ALE00500
ALF00510
ALE00520
ALEO0530
CALEO 540
LEOC550
ALEO0560
AXLE00570
-128-
SUoF(UiiN& A'r2.AN
!L1,L2,L3,PP,.JN1,JN2)
C
TdIS SUJ.A:~3UrIN C.!PUTrES liiE rI-ANSITIGN PROBABSILITY
?Cl? Al
A-;!1IC'N I'J Til3E OPEFA2i;ONAL STATUiS 0O. A MACIINr.:
C
P1I00760
PM10'770
d'109780
P.I3)3790
P."100800
----------------------------------------
PMI00820
PnISO3
'3
P I'-' 840
2PMIDU850
P1A' "860,
PMi110870
C
CC;,'!XON /?A::AAS/
AR[3),
AP(3),
NP(2)
C
INr 5F
i
L 1,L2,I.3,J1';l,JN2, Nlll, NN21,NP
C
F A L*15 .nA,P,AL3, PP
C
,
AL3 =L3 - 1
NN11 = NP(1) + 1
NN21 = :N?'2) + 1
IF
(L2 .;E.
1) 30 TO 10
10
1
5
4
il?1C88O
PIO00890
P11003900
PP = Ar :L1) *AL3 + 1. Q0-Ak 'L1))*
1. Q-AL3)
R ETU RN
IF 'L1-2) 1,2,3
IF (JN1 .EQ.
~NH11) GO TO 4
PP = A? (L1)* (1. Q0-?L3) + :1.0-AP'1))*IAL3
ET[E;
1
PP = AL3
ErnfiN
2 IF (JN1 .E.
GO TO 5
3 IF 'JN2-1)
END
1 .OR. JN2
4,4,5
.
N21) G
1..1
O O 4
.2:11)910
PM1100920
P13%,93;
PMI00940
? Mn!(950
PMI00960
P1100970
PMIo00980
PMI00990
PMI01000
P1Io101:3
P1I1020
PMI01030
PB1O1J#40
-129-
STI-CUJTIiNE A.T.NS (PECR, AP;'V, AFINAL, NI, NIP1, NIACH)
ATROO0010
C
ATE00020
-----------------------------------ATRO030
C--------------------------------c
T.i!S ST1j.CUTINE cCi2PUrES THE !'.OfDABILITT OF THE AACHINE STATE
ATO00040
C
T .A:SITIOCN OF?
ACHIN- 'N?.ACH' FiOM STATE 'APEV'
TO SAE
'AFINAL'ATOO050
C
ATFOCC60O
C
I'NI' ;AD
N'IP1' AFE 7TH? S-C'.AGE LEVELS OF THE UPSTREAM AND
£DCiNSST.5.A
BUFFER STCEAGES.
C
C
1'1AX' IS TH2 .MXIi.U¶. CAFACITIES OF IHE STORAGE AFEAT PADDED WITH
C
C
FICTIICUJS SIOF.AGES 1 AND K+1. TiJ FPOFMER IS NCN-E.IPTY AND
C
THE LATTER NGN-FUIL.
...................................
C--------------------------------------C
CC;
C,':/?AfRA.S/ f(3), P(3), N,¶AX(2).
iAX(4)
C
I !'T _r:, PY
AFINAL, NI, NIP1, NMACH
ATRCO070
ATPOCC80
ATFCUOS90
ATFOlC10C
ATi.90110
ATEROD12C
A
130
ATE00140
AT.FOO150
ATF00160
ATOC 170
'4N.AX,
IN-JE
kTF03180
~1AX
C
Ft--'* 16 PR.OB, QFLCAT
LZIAL*16 F, P
C
IF
C
C
C
C
C
(ARHEV .NE. O) GO TO 10
IF APPEV = 0 THE IACItlNE IS INITIALLY DOWN.
OF REPAIR IS INDEPENDENT OF THE LEVELS
TT'HEir -0P3.BILITY
IN ADJACT-NT STORAGES.
P]CB = yFLOAT(AFINAL) * R(Ni'ACH) +
QFLOAT(I
- AFINAL) * (!.C2+C
~EETU. N
*
-
P(NO!ACH))
C
13 CON INUE
c
C
I? TH. UPSTREAM STCRAG£ IS EMIPY OR THE DOiWNSTREAM STORAGE IS
PFULL, THEN THE PRCBABILITY OF FAILURiE IS ZERO.
OTHE.uISE, THE PBOBADILITY IS COMPUTED.
C
C
IF (NI .Q. 0 .OR. NIP?
.EZ.
%AX(NMACH+ 1))
PEiB = QFLOAT(1-AFINAL) * P (NMACH) +
*
QFLOAT(AFINAL)
*
(1.CQ+C
P(NMACH))
FETURN
C
20 CCONTINUE
PrCB = QFLCAT(AFINAL)
fETU.N
C
C ***** END OF ATIEANS
i:D
END
~ATI
GO TO 20
ATFOC 190
ATIFOC200
ATRO0210
ATRO0O220
ATRO00230
ATE002240
ATRO0250
ATOO0260
ATE 00270
ATROO280
ATROC290
ATE.00300
ATROO310
ATP00320
ATF30330
AT00340
ATR00350
ATROC360
ATRO0370
ATROO380
ATRi00390
ATR00400
ATROQ410
ATR00420
ATOO4 30
ATfR0044O
ATFOC450
ATR0D460
ATE00470
ATO0080
00490
ATOO50050
-130-
30U03910
B30U00320
030
aou3G04C
OIUN!S (X1iiAX,X2dAX})
SU3BOU7INE
C
C--------------------------------------C
THIS SUBOUIIINi CCavPUNES THE UPPER BCU:NDS
AND) 'X2AXI C:N 'XI1 ALD 'X2e ,SPECTLIVLY.
C
'X1l.XI
"*AX(LIMiTS)
C
BOGUND = 2.0
BCU1O03050
30U3O06C
70
C -------------------------------------
C
CO¢.S
MC/?AP X$S'/ (3) , P (3) ,V MAX (4
C.QON/F.ACTT-CF/.r 1,9 2 ,A 3,?1,?,3,Ci?
C
Lk 1pIC., 2P2.C~l 3 P3, h! fi2,
*
1,3,CU1i;2,
f: 3,i
,?.1,I0.P2,C1.P3,
3.
E 1C2,F !P3,h 23,i\2P1
t4O
.
A 1OUX
IiNTEGE:, bX, 1,NC2,)31C5121,N12,;~2f2, 2 IFF
C
3-u,315C
C
fiLAL1b
r AL'*16
*
*
C
c
C
U
1 LU E2,?Ut 3,
,P
l,,X%X 2ZA
, A,
i.1, I ,}3,P1tP2,P3,Cir, 1C!;Ei2,CGi
.
3P3, f1F2, ii15 3,
O.IR 1P 1,C Z:}.2P2,G
h 2 F 1, 2',F2P3, 3 P!, 3P2, 3P 3
CCO'.PU.-' UPP;tL
DU4 #4i AX
'"
3,Ci21 ,-.P2,CP3,
.. 2R3
P2, I1 P3
i 1P1 , k;1
t,P2P3,P2P23
,3
BCUND CH X1
-3030
t1
3P3)/ (O~P21'O.'I 323* (O.".Al *O.u.jD U 1 =Ci,3 * (CŽ.ii3-CSp2 *.iP 1*C
0.' 2*CIh3?P3*CGl1P1))
*
D U..2=0O P2/ (J.'M 2P2*03ofR3)
1))
/ C,1E 1)
(LO2*(D[J!2-OlF:.
DU.= 2
? 3*.Pl *O.5 2iP2)/(G0irP3*Cr, 2P2* (GIC 1*cIt2iL
,U3=OZ,. 2* (Or. 2-CL3
P2
.2*C 1P1 ))
=J1P3 "O I
E2* C" 3P3)
D U1 4=OP 3/ (c, .
CCOPUTE dPPEE
3CUND GU X2
2?22 (OER2*C,1R3*;i
DU a1=.Ih2 * (C3 2-C.fP31*C MP3*C.'h2P 2) / (OMcP 1*
CNP 1*CoIh 2P2*0b3P3 )
DU1 =1. .I+0O/DU
2OU)000390
DUv 2= CP 1/( C. 1P1*C0'i2)
D1U.%2 = Eu.i ( cu . 2 -CM.E 3 ) / (OPP3* iU."2-cOl 3P3)
*
1-C rP2*C MPJ*CMI-1P 1) / (C EP2* C'i-1El
i* ({O-.l323)
C: P 2C .F 1 P1 *CO._".
DU Y,3=01i.
"*
DUM3=1.0j+3/DJfE3
DU ,4=0o 012/ (C .P, 1 *C.2P2)
u4-CERU3) / (OMP3*L:U154-C8IE3P3)
D)U5i4=)DU.'1* ( U!
X2.AX =2.GQ+) * QmiAX 1 ()DUl,DUM2,LLu3,L0ri4)
C
E TUF.N
C
C ****
LND OF BCUNIDS
C
END
,
33OUg0163
30B'O i70
30U.u0-td3
CU31930
UJ00230
B~OUO0210
BOO;1J220
230
O U0C 240
3C1090250
DOU0O260
B(0,'I*DU!2BCO'03270
33U002803
BOU3 0290
BOU00 30
3ao000310
i0U3U032C
D U1-4= (0o P 1 *DU 3 4- Cl,. 1 1) / (oUMI*I ( D1UM 4-CNR 1 ) )
U2,1U.i3,DU!4)
A,1AX1 ('JUNl,I,
X1lAX=2. 5Q+3 *
C
C
c
30 1.O
3a8C
3e93090e
30030190
B
iiOU6C110
UtO,
(GCl1 *OJh 3-
30UJ3O330
3000) 34C
30UU335C
BOU00360
O3U00370
3OU00380
BO
BOU300400
BOUo.;
10
BiCUO 420
0ou0043C
3OU00440
0OU 0450
BOU03460
BO000470
--BOU0048C
s3CU03490
30U005CC
B30U00510
30000520
-131-
C
.
C
C
-
--
---
--
-------------------------------
-
SUB ROUTINE
B.IERRR.V1}
EX rI KN! ,,M,N,A,W,IP,
C
INErGER
I
EL,
M, N,II, IP, I 1,K K 1, LLL
,J, 1, a 1 , avI TS,IY-RR
IMlOol 0
-X 0
20
i xi 0;30
_VX M.GC040
_7X.ioo050
i.X~360
:X00o070
-_X 0080
z.X 00090
ELXMju103
C
c*4*****************
********$***t********t****$$**t*$$****J8*t***********X
30u 110
C
THIS SUBROUTINE IS A RrAL*16 VERSICN OF THE EISPACK SUBiOUTINE
zXi9Ov123
zX do0 130
C
MINFIT.
C1*******$***#************#** *************************
*****************Xi*J*4*
Xi.X0150
C
THIS SU3FROUTINE IS A TRAhSLATION OF THE ALGOL PROCEDURLE MINFIT,
Xz00O160
C
NUM. MATH. 14, 403-420(1970) 'Y GOIUB AND REl.NSCM.
ELX,3C170
134-151(1971).
C
HANDBOOK FOR AUTO. CCMP., VCL II-LINEAE ALGEBRA,
tX,30 180
C
.X10 0190
THIS SUBiROUTINE DETERMIRES, TOWARDS THZ SOLUTION OF THE LINEAR
C
T
rXMC0200
C
OF A P£XA213
REAL
C
SYSTEM AX=B, TlE SINGULAR VALUE DECO.POSITION A=USV
C
T
z.M00220
M B3Y N RECTANGULAR MATRIX, FORMING U B RATtiHER TAN U. HCOUS-HOLCBEzXM.o~230
C
iEM00240
C
BIDIAGONALIZATION AND A VARIANI OF THE Qk ALGORITHM ARE USID.
C
EXdOU250
C
ON INPUT:
X 00260
LX1M00270
C
C
NM MUST BE SET TO THE RON DIMENSION OF TgO-DISENSION AL
EXM0(,280
LX100o290
C
ARRAY PABAMETERS AS DEClARED IN THE CALLING P.OGRAN
C
DIMENSION STATEMENT.
NOTE THAT Nn MUST BE AI LEAST
LXMOU330
C
AS LARGE AS THE MAXIMUM OF n AND N;
EX M00310
EXM30320
C
C
M IS THE NUMBER CF RiCS CF a AND B;
.X00330
6XM00 340
C
£LXM0350
N IS THE NUMBER OF CCLUMNS OF A AND THE ORDER OF V;
C
51"00360
C
EXM00370
C
A CONTAINS THE RECTANGULAR COEFFICIENT MATRIX CF THE SYSTER;
C
LXMO00380
EXM00390
IP CAN BE ZERO;
C
IP IS THE NUMBER C? COLUMNS OF B.
C
XMOCG400
C
B CONTAINS THE CONSTANT CCLUN M.ATRIX OF THE SYSTE!
EXM00410
C
IF IP IS NCT ZERC. CTIHE.WISE B IS NOT EFERENCED.
LXM30420
EX-M00430
C
C
ON OUTPUT:
EXMCs4I0
C
EXI00450
XMSOu463
C
A HAS BEEN OVERWRITTEN BY THE MATEIX V (OETHOGOIAL) OF THE
C
DECCMPOSITION IN ITS FIRST N EOWS AND COLUMNS.
IF AN
EXMOO473
ZX33Ui480
ERROR EXIT IS MADE, TlE COLUMNS CF V CCRRESPONDING TO
C
i.X 00 490
C
INDICES C, CORRECT SINGULAR VALUES SHOULD EE CCERECT;
C
IX00(500
LXMO0513
W CONTAINS THE N (NON-NEGATIVE) SINGULAR VALU2S OF A (THP.
C
EIX00520
THEY ARE UNORBDRED. IF AN
C
DIAGONAL ELEMENIS OF S).
EXM00530
ERROR EXIT IS MADE, THE SINGULAR VALUES SHOULD EE CORLEECT
C
LXHOO540
FOa INLICES IEin+1,IERR+2,...,N;
C
EX5;)550
C
(N)
EAL* 16 A (NM, N),W{(N), E(N ,IP) ,V1
REAL* 16 C,F,G,H,S,X,Y,Z ,'PS,SCALE, rACHiLP
EEAL* 16 QSQiT,QiAX1, ABS, QSIGN
-132C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C ** **
T
B HAS BiEN OVERWRIT2EN BY U B.
IF AN ZEEOR EXIT IS MADL,
T
TH E RO'S Cl U B COF!ESPGiIINlG TO I!'DICES OF CORh.ECT
S.lGULAR VALUES SHGULD BE COfLECT;
IEFi
IS SEI 10
ZERO
FOE NCA.
AL RETUNF,
K
IF THE K-TH SINGULAk. VALUE FAS NOT BEEN
DETEEMi E:;D AFTEE 30 I TEFATIONS;
EVI IS A TEMPORARY STOEAGE AERAY.
'
QUESTICNS AND CCM.IENTS SIOULD BE TIFECrED TO B. S. GARBaO,
APPIIED MATHE;MATICS DIVISION, AhGCNNE NATIOCNAL LABOEATOEY
.X.
-
-
-
-
LXMCU563
.X 00570
EXm;S58)
tX1iOO590
L X1,O0600
X 0u613
;x.00 620
EX.3O630
2-XMOOb40
iXh_,65 3
LXf)3C660
XM003670
£X0J680
.-X.A30690
LX03700
X307 10
-72…
LX 30730
*ACiEP IS A nACHINE LEZPLND.NT PARAMIEP SP.ECIFYINGG
ZX:i007-0
THE RELATIVE PRECISICIN C;F FLOATING -POINT ARITIMETIC.
L.XJi750
ACHEP = 16. C0C** (-13) }OE LONG FORS AEITIHITIC
iXML00760
CN 'S360
..
.
.Xl0J770
X.O..:
*** ** ******ClIA NG D TO EXTEN DE PI.ECISIG************* **********LX
*0C780
DATA MACHE?/1.0OQ-32/
tX.0i790
C
:=X
00800
IERR
O
0=
EX 1OC81,3
C
::::::::::
HOUSEHOLDER EEDUCTION TO BIDIAGONAL FOES ::::::::::
-XM00820
G = O.0i_0
LXM"id33
SCALE = 0.OQO
ELX.08 40
* = 0.0'0
EX.00850
Z..X100860
C
DC 300 I = .1, 1
=X.t3i870
L = I + I
LXt 00 880
PVI (I) = SCALE * G
x 500890
G =
0%.0
X'U.O9090
S = 0.00Q
£X
t00910
SCALE = O. OQO
EX0-0920
IF (I .GrT.
M) GO TC 210
.-X100930
C
.EX00940
DO 120 K = I.
I
EXM30950
120
SCALE = SCALE + QABS (A (K,I))}X
00960
C
LXnZ36970
IF (SCALE .EQ. 0.0CO) GO TO 210
.X
MO3980
C
'
X'00990
DO 130 K = I, !
.xMO1000
A (K,I) = A (K,I) / SCALE
XO.010
1
S = S + A IK,I)**2
£301320
130
CCTINUB
.x
, 01030
C
EXH3010#O
FA ( II)
-.
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G = -QSIGN (QSQRT(S),F)
EXI 01060
B =F * G - S
L
3X81070
A(I,I} = F - G
£X131080
LXn)3109
IF (I ,EQ. N) GO TO 160.
C
Lxn31100
::::::::::
-133-
DC 150 J = L,
S = 0.0Q0
.
X
EX0 1110
-LX531120
-X
DO 140 K = I, MS
S = S + A (K,I) * A[l,J)
C
140
1X01130
XM01 140
LX£01150
EX -. O 1160
C
F = S/
HB
.XMA1170
c
''cX
150
150 K = I, J
A (K,J) = A (K,J)
CCNTIN;JE
160
IF
.l31180
EXn01190
.XM01200
LXiM1210
DO
+ F * A(K,I).
-
C
(IP
.EQ.
XM11220
0)' GO 10 190
Xi;X01230
ELX01240
C
LO
180 J
= 1,
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.IP
LxO 1250
s = 0.3OQ
EXJ)1260
iX £M01270
Ic
DO
170 K = I,
J
S = S + A (KI) * B{K,J)
170
c
F = s /
a
C
DO
I, J
= B (K,J)
:
180
180 K
B (K,J)
CCNTINUE
190
200
EC 2CC K = I,
a
A K,I) = SCALE * A{(,I)
210
w(I) = SCALE * g6
G =
.Q090
s = O.0U0
SCALE = 0,.000'
IF- (I .GT. S .OR.
+ P
*
I.K,)
.
C
C
I
EQ. N) GO 2IO 290
.X
t;01440
.
IXl 501450
XBi01460
EXtO 1470
C
220
DO 220 K = LI
SCALE = SCAL
U
+ CQAES ((I,K))
C
IF
(SCALE .E.
0.0Q00)
Exno1 480
ElMO51490
GO TO 290
C
EXO01500
EX 01510
LX£Y01520
CO 230 A1 L, I
A(.I,K)
A(I,K) / SCALS
S = S + A.(lK) **2
230
EXM01530
EX1101540
EXa31550
LZ.x01560
EXt501570
. Yx.01580
CCI TINUII
c
F = AfI L)
G = -QSIGII (QSQRT (S).
1 %* 6 - S
=
A(X,L) =6
)
LX01590
LxL.P01600
C
240
rO 240 K . L, I
111 (K) = A I(,K) /
C
IF
C
{I .EQ.
1)
GO TO 270
i:XM01280
LXJ01290
kX,2t01300
i.n:1310
E l01320
MX
XM01330
1xr,01340
r. XH01350
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x01380
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L xa01420
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50165X
O50
-134DG 260 J = L,
S = 0.00
I
-
z.X,101660
XMCt1670
LAX F.0 1680
ZXH/ 1690
£Xo01 700
-Exr101710
C
250
DO 250 K = L, N
S = S + A (J,K) * A(I,K)
c
c
260
DO 260 R = L, N
.£X01720
A J,K) = A (J,K) + S * RV1 (K)
CONTINUE
270
280
DO 280 K = L, N
A tI,K) = SCALE * A(I,K)
-XM01730
-:.
X 1 74 3
L.X1M31750
SXd31760
£LX:31 770
LXM 3 1780
X'31 790
EXg01800
;XMK01810
£X031820
LXM 31830
c
293
0
= Q2AX1 (X,QABS J (I)) + QABS(fV 1(I)))
300 CONTINUE
C
ACCU.'MULATION OF RIGHI-iHA!D TRANSEGRIATIORS.
C
FOP I=N STEP -1 UNTIL 1 LO -::::::::::
DO 400 II = 1, N
I =
* 1 - IIIF (I .EQ. N) GO- TO 390
IF (G .EQ. C.OQO) GO TC 360
C
DO 320 J = 1I, U
C
::::
: DOUBLE DIVI.SION AVCICS iLOSSIBLE UNDERFLOW ::::::::::
320
A (J,I)
(A {II,J) / A {I,L)) / G
C
·
;X
DO 350 J = L,
s = c.00o
C
:XMH01940
DO 340 K = L, N
3 40
S = S + A (,.K)
* A(K,J) '
C
DO 350 F = L, I
S * A(K, )
A(K,J.) = A(K,J)
350
CONTINUE
C
,.XM0020
360
DO 380 J =t,
IN
A -aO
(I,J) = 0.00
A(J,I)
O.OQO
380
CCNTINUE
C
390
A (I,I) = 1.0Q0
G = RV1(I)
L= I
400 COCNTINUE
C
0) GO TO 510
IF (H .GE. N .OR. IP .EQ.
51 = N 4 1
,.XMJ2130
C
DO 530 I = 111, 1
-X1.02150
C
.Xl102160
DO 500 J = 1, IP
B 1I,J)
0.oQO
500 CONTI NU
C
::::
DIAGONALIZATICN OF THE BIE1AGONAL FORM
:
£XEf0 1840
.
x101850
XLX101860
LXM01870
E1131880
fXEu1890
XM1 900
"l0
1910
-zXMO 1920
£iXM01930
iXMI01950
X1O01960
,.XI-j1970
LXn01980
XM01990
X 102
330
EXM 02010
2030
LX
LXM$2&W40
X 1102050
·
.
141X02060
EX M02070
i-X
132080
EXM02090
LX,02 100
EXH02110
£X 102120
.Xi102140
I1.102170
EXZ02180
£X B02190
::::::
£k
2203
-135510 EPS
= MACHEP .*
e
nX302210
-
' XH02220
-X102230
r02240
FOR K=N STfP -1 UNTIL 1 EO -- ::::::::::
DO 700 RK = 1, N
:EX
K1
N - KK
K = K1 + 1
ITS = 0
C
EST
E::::::::::
FOR SPLITTING.
C
FOB L-K STEP -1 UNTIL 1 DO -::::::::::
DC 530 LL = 1, K
520
C
LI = K -
C
C
530
C
540
kEX102250
LXM02260
XM.02270
XM62280
:.X!!2290
LL
LX~.92300
L = L 1 + 1X
IF (QA3S{RV1(L)) .IE. EPS) GC TO 565
R:::::::::V1(1)
IS ALWAYS ZERC, SO THiEiE IS NO EXIT
THROUGH THE POTTOM OF THE IOCP ::::::::::
IF [QABS(W (Ll))
.LE. EPS) GO TO 540
CCNTINUE
.X102360
1 ::::::::::
::::::::
CANCELLATICN OF ;VI(L) IF L GEATE THIBN 1XM02370
C = O. OQO
S = 1.OQO
c
DO 560 I = L, K
F =
· BV1 (I)
*
i 1({I) = C*
P.
V1 (I)
IF (QABS (F) .LE. EPS) GO TO 565
G = W(I)
H = QSQRT(F*FP+G*G)
w (I} = -HX
C = G / H
S = -P / I
0) GO TO 560
IF (IP .EQ.
'
.
C
0O 550 J = 1, IP
Y = E (L1,J)
Z = E(IJ)X
B(L1,J)
550
B (I,})
CCNIINOI
=
Y * C + Z * S
-Y
* S + Z * C
X1m02560
XY102570
EX102580
EX B02590
560
CCNTINUE
565
Z =
(K)
IF [L .EQ. K) GO TO 650
SHIFT FROM BOTTOM 2 BY 2 MINOR :::::::::
IF (ITS .E(.
30) GO TO 1000
ITS = ITS + 1
X = i (L)
Y =
(rK1)
G = RV1(KI1)
-EX02690
5 = EV1 (l)
F = ((Y - 'Z) * (Y + Z) + (G. - H) * (G * H)) / (2.000
G = CSQRT (F*F+1.+
G.O)
F
((X - Z} * (I + Z) + H * (Y / IF + QSIG (G,F)) NEXT QR TRANSFORMITION
C = 1.OQO
S = 1.OQO
C
C
TEST
FOR CONVERGENCE
-;:Xi02380
LXM02390
31.E02400
_XrZ02410
£XH02420
_X.Q02430
EXM02440
X 1102450
EXn02460
L 0 2 470
5X102480
XM02490
XM102500
EX1-02510
EXE02520
EX02530
E02540
EX502550
C
C
H12310
tXM-102320
EXM02330
'-X02340
£X-02350
ELX02600
:::::-::::
NO 2610
EX0
EX502620
EX 02630
Lx.02640
kYX
502650
EX802660
EXb02670
B X102680
*
H * Y)
H)) / -
EI*02700
EXO02710
i02720
L::::::::==
C2730
SXa02740
EX02750
-136C
DC 600 11 = L, K1
I = I!.
+1
G. = RVl (I)
Y = W(I)
H = S * G
.
G
C *G
Z = QSQRT (F*F+H*H)
.aVl(I 1) = Z
C = P / Z
S = H /Z
F = X * C+ G * S
G = -X
S + G * C
H = Y * S
gX02820
-
C
570
DO 570 J = 1,. N
X =. (J,I1)
z=
AIJ,I)
A (J,I1) = X * C + Z * -S
A (3,I) = -X * S + Z * C
C ONTI NO I
C
.C
580
Z = QSQRT (F*F+H*H)
.Xm33000
( I1) = Z
:::::::::: ROTATION CAN 3E A.BIT-RA.Y IF Z IS ZERO ::::::::::
IF (Z .EQ. O.OQC) GO TO 580
C = F / Z
S - H /
F = C *
S+ S * ·
X = -S *G
+ C
I
IF (IP .EQ. 0) GO TO 600
;LXM03070
C
590
CCNTINUE
C
Vl1 (1) = O.Co
RVI(K) = F
V (K) =X
GO TC 520
C
:::::::
CCNVERGENCE ::::::::::
650.
IF (Z '.SE. 0.OQO) GO TO 700
C
:::::::::
(K) IS MADE NCN-NEGATIVE
w(K) = -Z
C
-ro 690 J = 1, N
690
AIJ,K) = -(J,K)
C
700 CONTINUE
.
.~
EXM02830
EX 102840
EXa02850
X 2860
EXl'J2870
i-X02880
-XM?, 0 2890
£X 02910
EX702920
EXZ02930
LXI02940
EXZO2950
iEXfiJ2960
EX I 3297 0
EX902980.
z:*
XM02990
LXM33G10
kXI03020
XMu3C30
X 0 3040
XZ
EXMu3050
EXM03060
£-EXH03080
EXM103090
EX103100
EX,03110
EXM33120
EXM.03130
xnM3 140
Exn03150
L.X 803160
EXMU3170
EX
X 03180
EXM 03190
Exn03200
EXP103210
EXM03220
£XM03230
E::::::::::
XM032#0
'XM03250
EXM03260
DO 590 J = 1, IP.
-= E 11,J)
Z = B(1,3J)
B(I1,J) = Y * C + Z * S
B (I,J) = * S + 2 * C
CC NINUB
C
600
X
X02760
LX.02770
x32
XI
780
EX M,02790
gXZ028GO
2 8 10
£X nO03270
.
EX0i3280
.XH03290
EXXM03300
~~~~~~~~~~~~
~~~~~~
~~~
~~~
~~~
~~~
~~~
~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-137-Xi10331
GO TC 1001
'IO CONVERGENC I TO A
SET ERROR -:
:: : ::::
C
SI:'ULAR VALUE AFTER 30 TTERATIONS
C
1000 IER = K
1001 RETURN
LAST CARD OF MI';FIT ::::::::::
::::::::::
C
END
C
0
Ex M3 3 320
EX3,03330
X, 03340
X::::::::::
£.XM33350
iX E03360
EX103370
EX.03380
-138SUBROUTINE FORMBX
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C -C
C
C
C
C
C
C
C
C
C
(B, U, KSI, T, JCOL, ODSTAT, !1ODD, NKSI, NFODD) FROOE0010
FOR00020
-...
FOf.00030
THIS SUBEOUTINE FOPMS THE B MATfIX IN EQUATION B*C=O
FCOR00040
THE
B AATRIX IS THE CONDENSED FCRM OF .ATRIX
(T - I)*KSI
FOR00050
WHERE 'T' IS THE TRANSITION MATRIX AND
FOR00060
'KS1' IS THE MATRIX CCNSISrING OF KSI(STATE,UJ).
FORO0070
THE CONDENSED FOPM 'B'
ONLY ITNCLUDES THOSE POWS OF (T - I)*KSI
PORO00080
THAT ARE NOT IDENTICALLY ZERO.
FOG 00090
FORO0100
T' IS A SPAESE NKSI X NKSI 3ATRIX; 'KSI'. IS A DENSE NKSI X
FORO1.10
NCDD .MAThIX. THE CONDENSED MATFIX 'B'
IS NODD X NODD.
FOF0O120
FOE00130
SINCE 'KSI ' TAKES UP MUCH SPACE, IT IS GENERATED AND USED
FOPOO140
COLUMN BY COLUMN. THEREFORE 'B'
IS GENERXATD CNE COLUMN AT
FOR00150
A TI.E, AS THE RCOWS CF 'T'
ARE MULTIPLIED BY THE COLUMNS
POR00160
OF 'KSI'.
FCE00170
FOFC0180
IN 'FORMBX',
THE FIRST TWELVE COLUMNS OF 'KSI'
ARE EVALUATED
FOF03190
AT LI.ITING UJ (SEE SUBROUTINE 'LIMKSI').
FOP00200
FOR00210
'T' IS STOPED IN COMPACT F3RM, WITH ELEMENT T (I,J)
CORFESPONDING
FOR00220
TO THE ACTUAL ELEMENT T (ODSTAT(I), JCCL(I,J)) IN THE FULL
FOR00230
'T'
MATRIX.
FOR00240
FOR00250
THE SUBTRACTION CF THE IDENTITY MATFIX FROM THE TRANSITION
F0R00260
MATRIX IS ACHIEVED BY SUBTRACTING KSI(ODSTAT(I))
FROM EACH
FOR00270
PRODUCT 'ROW CF T'*'CCLUMN OF B'.
FOR0C280
FOR00290.
'U' IS THE ARRAY CONTAINING 'NODD' SOLUTION TO THE FIVE
FOR00300
INTERNAL TRANSITION EQUATIONS IN THE FOUR UNKNOWNS Xl1,2,Y1,Y2,Y3.FOB00310
FOR00320
'JCOL' CONTAINS THE COLUMN INDEX ·OF THE COERESPONDING 'T' ELEMENT.FOROC330
.
FOR 00340
'ODSTAT' CONTAINS THE fROW INDEX OF THE GIVEN 'ODD STATE'.
FOR00350
--- -------- -FOR
~ 00360
FOB00370
IN-EGER JCOL(NFODD, 11), ODSTAT(NODD)
F0R00380
INTEGER ICOL, NOD, NCDD, NKSI,NTELEM, IROW,JSUM,KSU
FOR00390
FOR 0000
BEAL*16 B(NFODD, NODD), T(NFODD, 11), KSI(NKSI),
U(NFODD, 5), SUM FOR00410
FOo00420
DO 121 ICOL = 1, NODD
FOR00430
PO00Q440
LOOP OVER THE COLUMNS OF THE B MATRIX
FOE00450
FOR00 460
IF (ICOL .LE.
12) CALL LIMKSI(KSI, NKSI, ICOL)
FOR00470
IF (ICOL .GT. 12) CALL FRMKSI(ICCL, NODD, NKSI, KSI, U, NFODD) FOR00480
DO 110 IRCW = 1, NODD
FOE.00490
FO00500
LOOP OVER ODD STATE BOWS
FOR 00510
COMPUTE B(IROW, ICOL)
FOR00520
FOR0053C
-139-
SUs = 0.OQ+O
NTELEN = JCCL(IROW,
C
C
C
NTELEM
IS
1)
+ I
TRE NUIIBEi O
NON-ZEFO
DO 100 KSUM = 2, NTELER
JSU. = JCCL(IRCU, KSUS)
SUM
SU= + T(ICW, KSUI)
CONTINUE
100
c
C
C
SUBTPACT CUT THE DIA.ONAL -1
110
120
C
C ****
C
B (IF.CW, ICCL) = SUN CONTINUE
.FOR00670
CC NTI NUE
RETUEI.
END OF FORMB
END
*
.ELE4ENTS IN
KSI(JSUM)-
* KS
KSI(CDSTl7(IROW))
THE POW OF T
FOE00540
FOR00550
F.CE000560
FOP00570
FOBO0 580
FOR00590
FPOR00600
FCP00610
FoP00620
F0OR006 30
PFOR00640
FOR00650
.FoR00660
FO 00680
F00OC690
FORO0700
FOR00710
FO 00720
FO00730
1-140-
SULEC'UTINE FPC.;PX(P,
C
C--C
C
C
C
C
C
C
C -.
C
C
C
C
C
C
C
C
C
C
C
C
U,
KS1,
C,
NODD,
NSTATE-,
SUMP, NFODD)
THIS SU3P0UTINE FOFMS THE NORMALIZES VECTOR 'P'
(TIlE
STEADY-STATE PROBABILITY VECTOR).
'P' IS GIVEN BY THE .MA'RIX EQUATION P= KSI * C
WEliE. 'KSI' IS NSTATE X NODD AND
'C' IS NODD X 1.
rC:IUSE 'KSI' IS A V:RY 'LAIEGE rMATRIX, IT IS ONLY FORiED-AND
USED ONE COLUMN AT A TIYE. 'P' IS FCrMED BY CCMPUTI'NG THE SU
CF C(J) * KSI'(*,J) CVEF THE COLUMINS OF 'KSI'.
'WSI' IS FORFED IN
'FP.
i:SI'K FOIMS THE
'INTKSI' FOFrtS THE
'LI'KSI
FORMS THE
THREE SPAFATE ROUTINES:
EXPFESSIONS CFRhESPCl;DING TO BOUNDAFY STATES.
EXFFSICNS CORrESPONDING TC INTEtNAL STATES.
EXPFESSIONS FCR THE LIdITING VALUES OF UJ.
'P' IS THEN NOR~MALIZED SC THAT ITS ELEtENTS SUM UP TC 1.
'C'
IS TIlE SOLUTICN CF B * C = 0
(i.E. THE WIGH7ING CCNSTANTS IN THE SUM EXPRESSION 'OR
…..--...……---…---
C
'PS).
FOE00240
C
INTEGEhi
NODD,N'STATE, NFODD,I,J
.
.
C
-iKAL*16 P(NSTATE),
C
C
C
-
U(NFODD, 5),
KSI(NSTATE),
C(NODD),
StUMP
IITIALIZE P AND KSI TO '
DUO
1
C
C
C
C
10 I = 1,NSTATE
P(I) = O. GQ+
KSI (I) = O.OQ+O
CCONTINUE
FO. EACH COLUMN FPOM THE FULL KSI THEN ADD TO THE ?ARTIAL P
SUMP KEEPS A P'NNING SU3 OF P
15
20
30
40,
SUMP = O. OQ+O
DO 49 I= 1,NDD
lF (I .LE. 12) CALL LIP.KSI(KSI, NSTATE, I)
IF (I .LE. 12) GO TC IS
CALL FFMKSI(I, NODD, NSTATE, KSI, U, NFODD)
CALL INTKSI(I, NODD, WSTATE, KSI, U, NFODD)
CONTINUE
DO 30 J =
,NSTATE
IF (KST(J) .EQ. 0..OQ+0) GO TO 20
P(J) = P(J) + C(I) * KSI(J)
SU.YP = SUMP + C(T) * KSI(J)
CONTINUE
CONTINUE.
CCNTI iZUE
FOOC3
o10
FOti00O20
F--------------------------------------------OO30
OP000040
F OF 0050
FOPOCC60
Foi00070
FOROCC80
FOPO0990
FOFOOlCO
FCRGIO110
FPCOF00120
PGEOC130
FO-RCC140
FORl00150
FOEOC160
FOh00170
FOROO180
ohOO3 1 90
FOFCO200
FOf G021.0
FORCC220
FOP00230
FCRO0250
CC260
FOR00270
FGE00280
F0R00290
FOROC 300
FC B00310
- 0 fi00320
FOR00330FQR00340
FOR00350
FOR00360
FOR00370
FOR00380
FOR00390
FORO0400
FOEOO00410
FORCC420
FOR00430
FOROC440
FOR0450'O
FORC0460
FOR00470
FOROO480
FCR004 90
FOR 00500
FORO05 10
FOR 00'520
F0R00530
c
C
C
:'C5.MALI7Z,
DC
50
PFOR0540
FChOSSO5
FOR03560
FCOOCSO0
FCFC?580
FOf00590
FOGOC6C0
FCR00610
P SO SUM EQUALS 1.
51 I = I,NSTATE
P(I) = P(I) / SU,3
CCNTINUe
~~~Ft~~~~~~~~~~UUP~
C
C
-*** .ND OF FOfGLP
FOP0620
FOR0630
c
END
El;DE
-CFOR00640
-142SUEJOUTINE FCI3.T(T,
C
C
C
c
C
C
C
C
C
C
C
C
C
C
C,
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
JCOL,
---------THIS SUIB:OUTINE! FORIS TtIF
ODSTAT,
NODD,
TRANSITION
NFPODD)
O........-----------M1AT1tIX 'T'.
FOR00010
FOR 00020
.000030
(NLY THOSE FCiS CF 'T' COPEES?ONDING TO THE 'ODD STATES' ARE
IFE2DFD. ?UhTiEIl;,ORE, 'T' IS VEiY SPARSS.
THUS, ONLY THE NONt.7-,.O
ELS:1ElTS OF THE r0CUS COEFESPONDING TO 'ODD STATES' APE
STCRED.
FCR FACH IOW THEFE APE NO MOFE THAN 10 ENTRIES.
FOt0 .ACH 'ODD STATE', A SEARCH IS MADE OVER ALL RECUFRENT
STATES TO DET-EMINE ;iiICR HAVE NON-ZEFO TEANSITION PROBABILITIES
TC HtiAT STATE.
TEE 'l'
MATRIX STCiiES THE
iTRANSITION PROBABILITIES, W-HILE THE
'C)STAT'
AND 'JCCL'
.ATRICES
CONTAIN TidE ROW A4D COLUMN
?CS1TiONS IN THE FULL TRANSITION MATRIX.
FOE EXAI?LE, TO FIND THE LDCATION OF T(I,J) IN THE FULL ?lATEIX,
'HE FI.C
INDEX IS ODSTAT(I), WHILE THE COLUMN INDEX IS JCOL(I,J) .
JCOL(I,1)
STLF3_S ThE NUMBER OF NON-ZERO ENTRIES IN THE ROWO.
CCUIENT STORAGE LEVELS AFE PASSED CN TO OTHER SUBROUTINES
THLOUGH COMON.
'NCPrEV' AND "'3?PEV' ARE DUMMYY STOPAGES: THE FORMER IS
h;ON'-E.PTY AND THE LATTER NON-FULL.
-FOOCC280
'N' AND 'ALPHA' D'NCTE IRE FINAL STATE (AN 'ODD' STATE).
'N-WN' AND 'IALPHA' DENOTE A OE3SSIBLE INITIAL STATE.
'-'1,'P2',AND 'P3.'
BARE TIHE MACHINE TFrANSITION PROBABILITIES.
'ALEGAL' IS A LOGICAL FUNCTION WHICH DETERMINES WHETHER A STATE
IS TFANSIENT (I.E. ZERO STEADY-STATE PROBABILITY) OB
RECUERENT.
FORC'J040
FOROOO50
PF000050
OFGFO0060
OR00070
FORC0080.
FCOO0090
FORCOlC0
FCROO1 10
FOR 00120
FO.00130
FOCiOO 140
FOPOO150
cFC00160
FOR00170
FOR00180
FCFOC190
FOC200
FOR00210
FO.00220
FlOR00230
FPC00240
FOR00250
FOPOC260
FORG0270
FOR00290
FOhCC300
FOfi00310
FOP00320
FOR00330
FPOOC340
FCOE00350
C----------------------------------------------------------------------FRO0360
C
CCMMON /PAhAMS/ R(3), P(3), NIAX(2), MAX(4)
CCI'.ON /STORES/ NOPREV, NPREV(2), N3PREV
C
INTEGER
INTEGER
INTEGER
INT EGEr
INTEG ER
JCOL(IFCODD, 11), ODSTAT(NODD)
N1KAX, IAX
NOPFEV, NPREV, N3PREV
I, ICNT,IN1,IN2,IA1,IA2,IA3
ALPHIA(3), IALPHA(3), NEWN(2), N(2),
C
E£AL*16 T(NFCDD, 11)
E-AL*16 R, P
RErAL*16 P1, P2, P3
C
'' FOEO500
LCGICAL ALEGAL
C
C
FOR00370
FOR00380
FOP00390
FOROO0400
FOR00410
FOR00420
FOE00430
FORO00440
IA1, IA2, IA3, INl,IN2FGOI00450
PFOR00460
FOROC470
FOR00480
FOE00490
_EGIN LOCP OVE
'ODD' STATES.
FCR00510
FOEOC520
FCE00530
-143C
DO 150 I = 1,NODD
CALL STOFh(ODSTAT(I),
ICNT = 1
30 143 IN1 = 1,3
C
C
C
CHECK
N, AL?BH)
NEI',;iBOEING STATES FOE POSSIaLE TRANiSITICNS OF N(1)
= N(1) + IN1 - 2
1F (Ni EV (1)
LT. 0 .01.
NPE7V (1)
DO 130 IN2 - 1,3
N REV(1)
FOECC620
GT.
NMnA (1))
F0O00630
FOOC0640
FOR00650
CHECK NI.-IGHd3.ING STATES FCE. TRANSITIONS OF N1(2)
FOR00660
FOC00670
IF (1N1 .EQ. IN2 .AND.
IN2 .NE. 2) GO TO 130
FOR00680
NPF-V(2) = N(2) + IN2 - 2
FOF03690
IF (N?r:EV(2) .LT. 0 .OR. NPEV(2) .GT. I'1AX(2)) GO TO 130FOR00700
FOR007 10
CALL N T'ANS(NPFEV(1),
N-'WN(1),
ALPHA(1), ALFHA(2),
1)
FCh.00720
IF (NEWN(1) .NE. N (1))
GO 0O 130
FOfiO0730
FOR00740
CALL ;TRANS (PfiEV (2),
NEWN(2), ALPHA(2), ALHA
, (3)
2)
FOF0O750
IF (NWNE(2) .NE. ; (2)) GO TO 130
FCE00760
FOE 00770
AT THIS ?GINT, NPREV(1) AND INPinEV(2) ARE POSSIBLE STORAGE
FOROC780
LEVELS WHICH HAVE NON-ZERO TPASITION PFOBABILITIES TO THE
FOR00790
FINAL LEVELS N(1) AND N (2).
FOP00800
FOR00810
IHE IACHINE STATE TRANSITIDNS ABE NOi CHECKED.
FOPOC820
P0B00830
DO 120 IAl = 1,2
FORCC840O
C
C
C
C
C
C
C
C
C
C
C
C
IALPHA(1)
= IA1 -
IALPHA(2)
*
J*
GO TO 140
I
FOE00850
CALL ATRANS(P1, IALPHA(1), ALPHA(1),
IF (P1 .EQ. C.OQ+O) GO TO 120
DO 110 IA2 = 1,2
C
C
C
C
C
C
= IA2 -
2, NPREV(1),
1)
1
CALL ATEA!S(P2, IALPHA(2), ALPHA(2), NPEV(1}),
NPEEV(2), 2)
IF (P2 .EQ. O.OQ+0) GO TO 110
DO 100 IA3 =- 1,2
IALPHA (3) = IA3 - 1
CALL kTRANS(P3, IALPHA(3),
ALPHA(3),
NPREV(2), 2, 3)
IF (P3 .EQ. O.0Q+0) GO TO 100
IF (.NOT. ALEGAL(NPREV, IALPHX))
GO TO 100
HEZE THE S'ATF (iPRE7V(1),NPREV(2),IALPHA(1),IALPHA(2),IALPBA(3))
WILL TJND-RGO A TRANSITION TO STATE (N(1) ,N(2),ALHA(1),ALPHA(2),
ALPH{(3)). WITH PFOBABILITY = P1*P2*P3.
THE APPF.OPEIATE ENTRIES ARE 3ADE -INTO 'T' AND 'JCOL .
ICNT = ICNT
1
T(I-, ICNT) = P1 * P2 * P3
JCOL(I,ICNiT) = NOST(NPREV, IALPHA)
C
C
C
FCR00540
FORCO550
FOE00560
FOR00570
FGR00580
FCE00590
FCE00600
FOR00610
END OF IALPHA(3)
LOOP
.
FOR00860
FOR00870
fOf00880
FFOR00890
FO00900
POEC09 10
FOR00920
FOROC9 30
FOR00940
FOROC950
FGR00960
FORC0970
FORO0980
FOR00990
GFOR01000
FPOo1010
FOE01020
FOR01030
FORI0104C
FEO01050
FORO1060
FOR01070
FORO1080
VOR01090
FORE01100
FCfiO110
-144-
lC
END OF IAL?HiA(2) LGOP
C
CONTIN14U
110
-
C
C
C
FND OF I.AL?IA (1)
END OF NPREV(2) LOOP
C-NTINCC NTINUE
END CF NPREV(1) LOOP
13C
C
130
C
C
C
CONTINUE
.FOR01290
£'ND OF LOOP CVTER ODD STA ES
140
C
C
C
JCL
150
**+**
LOOP
CCNTINUE
120
C
C
C
C
C
FOF01120
01130
P~~~~~~~~~~~~~~~~~~~~FCF.
FCF. G11 O
CONTINUE
C
(I,
= ICNT -
1)
END
FOR013C0
FOP01310
FOR 01320
FOB01330
FOR01340
FoaC!350
CCGTINUE
RE'URN
LND CF FOC'
1
FOR01160
FCF;01170
FOR01180
!90
OPOl
FCF01200
FOR 0121C
FOR01220
FOR01230
FOR01240
FOR01250
F0R01260
FOP01270
CFCG1280
T
FOE01360
FO01370
FOE01380
-145-
SUBROUTINE FCRMU(U, NODD, NFOLD)
C
C------------------------C
THIS SUBROUTINE GENERAIES 'NOCD' SOLUTIONS OF THE FIVE
C
PARAMETRIC EQUATIONS IN FOU.E UNKNCWNS.
C
THE SOLUTIONS AEE UJ = (X1J,X2J,Y1J,Y2J,Y3J).
C
C
THE EQUATIONS ARE REDUCED TO A SINGLE QUADRATIC EQUATION WITH
C
ONE INDL'?EHNTNT VARIABLE (X1 OR X2).
C
C
ACCEPTABLE SOLUTIONS ARE THOSE FCR WHICH X1, X2 ARE POSITIVE.
C
(OTHEitWISE THIE PROBABILITIES WOULD BE PEPRODIC WITH
C
PERIOD TWO IN STGPAGI LEVELS).
THIS IS CHiCKED IN SUBROUTINE 'UTRY'
C
C
C
THE SUBROUTINES CALLED TO SOLVE THE QUADRATIC EQUATIONS ARE
C
'Y1SOL', 'Y3SOL', 'ZSCL'.
C ---------------C
COMnON /PARAMS/ R(3), P(3), Na.AX(2)
COMMON /USOLV/ ALPHA(3),
BETA(3),
GANMA(3)
C
INTEGER NODD,NFODD,L,N.MAX,IIKAX,II,IX,N1,N2,I,J
INTEGER IFLAG,IFLAG1,LA,
AG2,IFLAG,IFLAG4,IGY1,FLG1ILGY3
C
REAL*16 U(NFODD, 5)
REAL*16 R, P
REAL*16 ALPHA,. BETA, GAMMA
REAL*16 Z(3,8), W(3,8), PHI(3,8), Yll, Y12, Y31, Y32,
X1MAX, X2'!AX, X1DIPF, X2LIFF, QFLOAT
C
L = 1
C
COMPUTE UPPER BOUNDS ON 'X1' AND 'X2'
C
C
AS WELL AS STEP SIZE.
C
CALL BOUNDS (X1MAX,X2NAX)
X1DIFF = X1IAX / QFLOAT(4 * NMAX(1))
X2DIFF = X2LAX / QFLOAT(4 * NMAX(2))
IIMAX = dAX0(4*NMAX(1),4*NMAX(2))
C
DO 200 II = 1,IIMAX
DO 190 IX = 1,2
X1 = X1MAX - X1DIFF * QFLOAT(II)
X2 = X2MAX - X2DIFP
* QFLOAT(II)
IF (IX .2EQ. 2) X1 = X1DIFY * QFLOAT(II)
IP (IX .EQ. 2) X2 = X2DIFF * QFLOAT(II)
C
N1 = 4
N2 = 4
C
CALL Y1SOL(X1, Y11, Y12, IFLGY1)
IF (IFIGY1 .LT. 0) GO TO 20
N1 = 1
C
FOR00010
P0F000020
FOPOO020
FOGO30
POfo0040
FOR00050
FPOS0060
FOE00070
FO00080
FO00090
FOR00 100
FOO00110
POE00120
FOR00130
FO00140
OP.
FOOU1 50
F0600160
FOR0017C
O------------------0180
FOE00190
FOR00230
FOR00210
FOR00220
FOR00230
FOR00240
FOR00250
FOR00260
F0R00270
FOR00280
FOfi00290
FOR00300
FOR00310
FOR00320
FOR00330
FOR00340
FOR00350
FOR00360
POR00370
FOR0t380
FOR00390
FOR00400
FOR00410
FOR00420
FOR00430
FOR00440
FOR00450
FOR00460
FOR00470
FOR00480
FOh3O490
FOR00500
FOR00510
POR00520
FOR00530
FOR00540
FOR00550
-146-
Z(1,1) = 1.04+O - .(1) + P(1)
Z (1, 21 = Z( 1, 1)
1. C+C - 5(1) + P(1)
Z (1,3)
~,3)
41=
Z(1,f4)
FO.00560
1FOJOu570
F'OiO05t0
* Y11
* Y12
.FCi)O590
P?3)26 C
? Os0f) 610
'lOR00620
oF0.E
0630
FOE.00640
?O3).03650
FOE-30 660
F03.00670
FCP3O680
FOFC'690
MOi)0700
'0.F33710
C
C
1,
CALL ZSOL (Z (1,1),
2,
3,
Z (3,1), Z (3,2),
IFLAG1)
C
1, 2,
CALL ZSCL(Z(1,3),
3
A..tD.
IF (IFLA.1 .LI.
3, Z(3,3), Z(3,4), IFLAG2)
O) GO TO 200
IFLAG2 .LJ.
C
20
CONTINUE
CALL Y33OL(X2, Y31, Y32, IFLGY3)
IF' (1:-LY3 .LT. 0 .AND. IfiGY1 .LT. C) GO TC 200
iF (1-'L,3 .LI. 3) GO .IO 30
C
Z(3,5) = '.cQ+C
( 3, 5)
Z(3,6) =
-
5(3) + E;(3)
POE00720
* Y31
PO0O37 30
FO 0 74C
FG33750
FOB0Si760
?CIF300770
FOR0O780
7FOEC0790
P0113800
FOECCoSl
FC 103820
FOh'0830
FCE00840
C
Z(3,7)
Z(3,8)
=
*=
1. 3Q+0 - a(3) + P(3)
Z(3,7)
* Y32
C
CALL ZSOL(Z (3,5). 3, 2, 1. Z(1 5),
Z(1,b),
IFLAG3)
C
CALL ZSCL(Z(3,7), 3, 2, 1, Z (1,7), Z(1,8), IFLAG4)
IF (IFLAG3 .LT. 0 .AND. IFLAG4 .LI. O) GO TC 30
tC
N2 = 8
CGNTINUE
33
?·70110v850
C
DO 130 I=Pl1, N2
Z(2,1) = 1.0.+30 /
(Z(1,I) * Z(3,I))
C
i)0 93 J=1, 3
W(J,I) =
U(L,2+J)
PHI (J,I)
CONTI NDIE
U(L, 1) = PII
U(L,2) = PI
90
ALI.A(J) * (Z(J,I)-3BTA(J)) / (Z(J,I)-GA'llA(J))
= (Z(J,I) - 1.CQ+0 + 2 (J)) / P(J)
/ Z(J,I)
= W (J,I)
(1,
(1,
I)
I)
* PiI(2,
I)}
C
IF
(I._Ej.1
YXIMAX,
IF (I.E2.3
X1iAtX,
*
IF (I.-Q.5
X1!lMA,
IF (I.£Q.7
' X1YAX,
"
IF (L .GT.
CONTINUE
1C0
190 CONTINUE
200 CONTINUE.
300 CCNTINUE
*
C
.OR. I.EQ.2)
X2MAX, L)
.0R. I.QZ.4)
X2!MX, L)
.0h. I. EQ.6)
X2MAX, L)
.OR. I.ZQ.8)
X2MAX, L)
NODD) GO TO
U(L,2),
IFLAG1,
CALL UlhY (U (L,1), U(L,2),
IFLAG2,
CALL UORY(U(L,1),
U(L,2),
IFLAG3,
CALL UiiY(U(L,1),
U(L,2),
IFLAG4,
CALL UTfiY(U(L,1),
330
Ff130360
FO0O0870
FOP0C0880
OF100890
F0130900
70500910
FO.00920
0FEOO930
FO003o940
0ii00950
PFOR00960
FO1i00970
FiO00980
0a.o00990
701101000
F0101010
EOE01020
OPI01030
PFO01040
FOR01050
FCR31060
0E.
OE1070
FOh
01080
7F0111090
FOE01100
-147-
BETURN
C ***** END OF FO[RlU
C
'
,
END
-
PFC!112
FOEgO 1 120
FOF.O1130
PFOEO1 110
PFOR01150
-148SUiHlOUI'Jr
E FiK-SI (tilELE.I, NDIMU, VKSI, KSI, U, NFODD)
C
·
C --------------------------------------.
C
THiIS SJBEO.UrINE CCLPUTZS THE NCN-ZERC ELEMENTS OF THE KSI
C
VEYCTOE
FOR /IiE GIVEN U(lUELEM) WHICH IS THE SOLUTION
C
(X1 X2 Y1 Y2 Y3) Ci THEi PARA..TE.iC EQUATIONS.
C
P
S uJ.(
C
*
KS1( O ).. )
FRO00010
FR00020
00030
Fis'00040
FRIMO050
FROCC060
?R3300;O
C
FR.MCCO8C
J
J
J
C----- - - - ---------------------------------------------------C
CO'inCA /3AEAMIS/ ; (3), P(3), NMAX(2)
COit:iCN /FACtCh/ Fl,
3,2,P13,
1,, P2, P3, C1,
i2, OMR3, GMP1,
*
CMP2, 0:P3, CS121, O1.[2P2, OhMF3P3, I.1F.2, .1E.3, 51?1, R1P2,
*
E1P3, P2E3, R2P1, 1,2P2, R2P3, R3P1, fr3P2, R3P3, P1P2, PP1E3,
*
?22P3, N1, N2, N1.ii,
i2M1, N1112, N2.12, NDIF
C
INTE GEi NMAX,I,J,iUEtLE
K.
S,
NK
I D NL IMU,NKS1,N
,
1,N2,N1,1 ,N2
1 1,12, N2M2, NDIFF
INT EG ER
C
B EAL*lb R, P .
REAL*16 KSI(NKS1),
U (NFCDD,5)
REAL*16 X12(100), X2P(100),
* il,I2, ri3,
P1, P2, P3, Xl, X2, Y1, Y2, Y3, ZI, Z2, Z3,
* OMFF3, C.i2
OrF3, , C£1, CM P2, CmP3, OE.51P1, C'1P2P2, OAit3P3,
*.R1f2, E.133, R?11, ,1P2,
R1P3, B2R3, [2r1, B2P2, E2P3,
* E3D1, R3P2, E3P3, 21P2, 21P3, P2P3,.
- X1X2, X1Y1, X1Y2, Y1Y3, X2Y1, XiY2, x2Y3, Y1Y2, Y1Y3, Y2Y3,
* FACT
C
C
FOR CCGDPUTATIGNAL £FFICIENCY AND SIMPLIFYING TrH
LXPRESSIONS A
C
NUIHBSIiUF CCMaCE LXPEESSICNS ARE CALCULATED.
C'
C
X1 X2 Y1 Y2 Y3 = U (,1)
U(I,2) U(1,3) U(1,4) U(1,5)
I=IUELER
C
C
EI = R (I)
1=1, 2,3
C
PI
P (1)
1=1,2,3
C
C
NI = N-AX(I)
I=1,2
C
·
C
X1P(I): X1 ** I
C
X2P(I): X2 ** I
C
C
OMRI: 1.0+0
R(I) I=1,2,3
C
0HPI: 1.'3J+0 - P(I) *=1,2,3
C
o0iR12l: 1.0U+3 - P(I) ,(I) 1=1,2,3
C
C
XIXJ, XIYK, YIYJ: Xi * XJ, XI * ¥YK,
I * YJ
FOR ALL I,K J>I
C
C
rIfJ,
RIPK, PIPJ: .El
F.J, El * PK, PI * PJ
FOE ALL I,K J>I
C
YI
FOR I=1,2,3
C
ZI = 1.0Q+O - '1 + 21P
C
FOE 1=1,2
C
NI.1, NI12: N1 -1, NI - 2
C
C
'NDIFF ]=8 ' (N2 +1) hiiICH IS THlE KSI VECTCR ADDRESS DIP-
.
F RM0 1.30
FR.'OO110
F500120
FRM30130
?R'ioi140
FPr.iC150
F1.;10)160
FE-&V
;170
FRMOC180
?PMI3C190
RM00200
- FR5?n1210
FR.MC0220
FERCO230
FRM?0240
FRrO0250
FE r,00260
FRMC0270
FRM00280
FEM00290
FRC03o300
ERM00310
FRMCO320
FRMOC330
FR MOO 340
FR100350
PRMCO 360
FP.MC0370
PRO00380
FRM00390
PR00430
PfiMO0410
FE100420
FRMOO 13.0
FPMCC4O40
PFRQ0450
FiRCO460
FEM00470
FPREOG48C
FE.CC490
FPE 03500
FRP.00510
FE M00520
FPMG0530
FRMOS054
P.:M04550
-149-
FE-'.'gCEv UiETWEEN STATES
C
(1 J K L 1M) AND
(i+1 J K L n)
FR1G057dO
C
C
C
-?R1C590
BEGIN ?RhE-CALCULATION OF FACTCR$S
X1
X2
Y1
Y2
Y3
=
=
=
=
=
FR103690
FRC 6 10
FH.-P00620
FM0 36 30
FCJ6.4C
FRA.3)650
1FPRt'660
F R 3670
F 'i
33630
F Y.00690
?Ej7)0
FP30)710
C 0,7 20
FFE. i73O
`0750
FRMC00750
FRM03760
FR i0770
U(IUELEL,
1)
U (IUELLi,2)
iUE(iUZLZ , 3)U(IUELEM,4U
)
U(IULLE.M,5)
C
FACT = 1.0Q+0
DO 10 1=1,N1
FACT = FACT * X1
XlP(I) = FACIT
CONX2NUE
10
.
C
C
.?i
FACT = 1.OQ+0
DO 20F
20 i-1,N2
?AC£ = FACT * X2
X2?(I) = FACT
CCNTINUE
20
-FR.10780
X1X2 = Xl * X2
X1Y1
= X1 *F?,C0800
* Y1
:1Y2 = X1 * Y2
YFHMCC820
XlY3 = Xl * Y3
X2Y1 = X2 * Y1
X2Y2 = X2 * Y2
X2Y3 = X2 * Y3
Y1Y2 = Y1 * Y2
Y1Y3 = Y1 * Y3
Y2Y3 = Y2 * Y3
FRMC3790
.
F
P
?RE00860
FfMCG870.
?RmG0380
FRAo03890
Z1 = 1.0Q+0 - 31 + P1 * Y1
Z2 = 1.0Q+3 - a2 + P2 * Y2
Z3 = 1.0+*0 - B3 + P3 * Y3
C
C
C
c
C
c
FR?,O0900
BEGIN COMIPUTING KSI VECTOR IN APPF.QXI.IATELY OEDiE
0
0 1 1 , 0 1 0 1
,
A3D
D
0 1 0
CF KSI(1)
1
= XlX2 * Y1Y2 * COR1 * (R1 + f3 - 1.123 - R1P3) /
(1P3 * R1)
.Fmot010O
KSi(11) = X1X2 * Y1Y2 * O0a11 / E1
KSI (12) = X1X2
Y112 * (F.1 + R3 - R1R3) / R1P3
KSI(4)
*
3810
FRIIGO830
PFRI'C0840
FRM00850
c
C
C
C
C
PR.10
O560
.Ff101040
0 2 0 1 0 TO 0 N2-1 0 1 0
FACT = X11l * (CdE2 * X2 / Z3 - OGP1 * 0fR2P2) /
J '= 19
DO 103 1 = 2,N231
KSL(J)
i=
ACT 4 X12P(l-1)
ElP2
FRi00910
FR°00920
F.M00930
PFRM0094O
ON UPFRMC0950
?FR.CO960
1, 970
F6FR00980
?RECO09.90
FRIC1000
F?!01020
F.M31030
PRM01050
FFRN01 060
FEO3 1070
FRM31080
FRM01090
FFRC1 lC
-150-
3
J · 1
KSI(J) = rAC'
J= J
7.
100 CONTIN3E
C
C
C
* Y3
*
I301110
F,.0C1120
FR. 01130
F. V1140
FE HOC
1150
1160
1F01170
FiY01 130
X2?{I-1)
0N2 0 1 0FR[.C
-R:':O
J = N2
* 8 + 3
KSI (J)
= Xl *
(P2
*
C
C
C
4
X2P(I;2w1)
*
(({CAP2
- OBi2P2
$ GMR1
* CMI3)
FR,0121C
FOR A =
0 0 ,
0 3 1i
1 1,
1 1 0,
* -Y3
* Y2 * Z3 /
P3
* Y1.Y2
+ Y1
N2-1 0 0 1
AND
*
*
(CO1A2
Z3 + P223 * Y2-Y3)
/
1 1,. 11 0, . 1
I
0 1 0, 0
1 N2-1
0 1
J = NDIFF + 8 * N2.1 + 1
KSI(J+1) - O.OQ0+
KSI(J+3) = X1Y3 * X2P(N211)
C
C
C
1 N2 0 1 0
AND
FF;:01230
FRMC 1240
FRn101250
FR'10 1260
F?R01270
FEO 1230
FBR0129C
FiM01 30
. FRi01310
FR f01320
FRMO1330
.FR"101340
1 2 A TO 1 N2-2 A
FOR A -=
0 3, 0 3 1,
1
1 1 1
.
J = N-IFF + 17
DO 113 I = 2,N2111
KSI(J) = Xl * X2P(I)
KSI(J+l} = X1Y3 * X2P(I)
KSI(J+2) = X1Y2 * X2P(I)'
KSI(J+3) = X1Y2 * Y3 * X2P(I)
KSI(J+6) = XlY1t
X2P(I) * Z2 / P2
KSI(J+7) = XlY1 * Y3 * X2P(I) * Z2 /
J = J + 8
110
CCNTINUE
C
C
C
1P3
.
1 1 A
J = t;DIFF + 9
KSI(J) = X1X2
J = J + 1
KSI(J) = X1X2
3 =3 + 2
KSI(J) = X1X2
J - J + 3
KSI(J) = X1X2
J = J + 1
RS1 (J) = XlXr
c
C
C
C
FRM01190
FR M0122C
O 3 1 AiD 1 0 1 1 1RMUC122C
J = ND.FF + 2
KSI (J) = XIX2 * (31I'1
* P2 * Y1Y2 /
1 + CP33 *
ClR1 * Ci%2) / (°3 * (El +
* R2RlfE2))
KSI(3J+)
= X1(2 * Y1Y2 * (E1 + i3 - r1.l3
1P3) ./
C
C
C
/
{(1 + i3 - E1R3)))
1 ARF
SPECIAL CASES
Z*
2 / P2
1 N2 1 1 0
3 = D1lFF + 8 * N2 + 1
KSI(3J+2) = X1Y3 * X22(N2M1)
P2
P2P3
FI101350
FEiMC136
FRB01370
FRM01380
FRM01390
FRM01400
PRM01410
F
FR101420
FRF101430
FR-101 4140
FR"01450
FRF01460
FRf.01470
FRMOt1480
FR.FO01490
FRM01530
FR101510
FRM01520
FM1101530
FRM01540
FRM01550
FEM01560
FR
F01570
FRM01580
FRSi01590
FPM01630
FH1'01610
FPPM01620
FRM01630
FFM01640
*
(01FR2 /
Z1
FPR01650
-151-
*
- X1
*
CXb2P2 * C:P3)
KSI J+6) = KS (J+2)
c
C
C
DO
/
?PRl 1660
h3P2
$ Y1
LOCOP? OViiA TtiE INTLiNAL STATES OF THE N1 STOR(AGE
.
DO 120 I = 2,N1.2
C
C
C
I
0 03 1I A;3D I
J
-?RfG1710
0 1 0 1
F + 1
KSIl J+1) = X2Y2
X1P(I-1) *(OMB3 /
X
/ 2PE3
KSi J+5) = Y1 * KSI (J+1)
*
C
C
= I * NDI
I 1 A
FRt01670
RO1F680
FEC1690
FRRO 1730
FOR .A= 0 0,
Z1 -
0 0 1, 0 1 1, 1 0 0,
X1
O.'MP2
1 0 1,1
' o.i3P3)
1
c
.
J = I * NDIFF + 9
KSI (J)
X1(1) *
KSi(JJ1)
- XlP(1)
RSI(J+3) = X1P (I)
KS J'+44) = X1(1)}
KSI J+5) = XI (I)
KSI(J+7) = XIP(i)
C
c:
I N2-1 A
FOR A = C C 0,
J = L * NDIFF +
FAC: = X1P(I) *
KSI (J) = FACT
KSI (J+2} = FACT
KSI(J+3) = FACT
KSI(J+4) = FACT
RSI (J+6) = FACT
KSI(J+7) = FACT
C
C
C
I
N2 C 1 O
AND I
C 1 0,
N2.1 * 8
X2P(N231)
3
1 1,
P3
1 0 0, 1 0 1, 1 1 1
1
* Y2
* Y3 * Z2 / P2'
* Y1
* Y1Y2
* Y1Y3 * Z2 / P2
N2 1 1 0
J = I * NDIFF + N2 * 8 + 1
KSI(J+2) = XlP(I) * 12P(N21!1) * 13 *
(.(OER2 / Z - Xl * 0,.1I2P2 * COMP3) /
KSI (J+6) = KSI(J+2) * Y1
*
C
C
C
X2
* X2Y3
*MX2Y2 * Z3 / P3
* X2Y1
* X2Y1
1
Y3
, X2Y1
Y2 * Z3 /
R3P2
'.
END OF LOOP OFP I
CGVE N1 STOIAGE INTERNAL STATES
12C CONTINUE
c
C
C
N1-i
0 0 O1-1
0 1 AND
C 1 0 1
J = NlMl * NDIFF + 1
KSI(J+1)
X=1
XP(N1I2)
* X2Y2 *
*
(CMR3 / Zl - X1 * OP2 * (.,R3P3)
KS (J+5) = KSI(JO+1) * Y1
C
C
C
N-1i
1 A
FC
A =0
0 O,
C 0 1,
/
1 C C,
i2P3
1 C 1,
1 1 1
?RM0O1720
?F0 1730
?FR 01740
FRx01750
FE. M 1760
ROQ1 770
FR;1nl7dG
FRM01790
IhHOC1800
V .018a1
Fm
a
FRM01820
101830
R.
FRM01840.
FRMi1850
FRMOd60
FRM01870
FRMO183O
FEM0 1890
FRO1 930
FRr01910
fP RM01920
FRM01930
FRMC1940
FR 0()1950
FRo01960
FRM0 1970
FRNM 1930
. FRP01930
FR1M02000
FRPM2010
FBR?02020
FMC2030
FRM02040
FRM02050
PFRM02060
FRM02070
FRE02080
FRa02090
PRN02110
FR!L02110
FP..C2120
FR.02130
FR.02140
FRI02150
FEMO2160
FER02170
FRM02180
?RMC219C
FR M02200
-152-
J = tl1,1
KSI(J) =
KSI (J+)
KSI (J+4)
KSI (J+5)
KSI(J+7}
C
C
C
C
* NlFF' + 9.
XlP((N1) * X2
= X1P(:?1X1) + X2Y3
= X1P(Nlitl)}
X2r1
= Xli?(N11;)
* X2Y1 * Y3
= XlP ({131)}
X2Y2 * ZL1
a1-1 2 A TO N1-1 ,42-2 A
FOP A = G 0 C, 0 0 1, 1
0
1 00
J = N1:11 ,
:DIFF + 17
DO 130 I = 2,N1252
FACT = X1P(NlI.)
* X22(l).
KSI (J) = FACT
KSI(J+1) = FACT * Y3
KSI(J+4) = }ACT · Yl
KSI(J+5) = FACT
Z3 / P1P3"
-FR02270
,
1 1 0,, 1 1
.
* Y1Y3
KSI(J+6)
FACT ' Y2 ' Z1 /
KSI(J+7) = K.SI(J+6) * Y3
J = J + 8
130 CONTINUE
C
C
C
-
.
'11-1 N2-1 A FOP. A = 0 0 0, 0 1 1, 1 0 0,
= N1131 * NDIFF * N2a1
* 8 + 1.
FACT = X1P(NlMI)
* X2P(N2M1)
KSI(J) = FACT
KSL (J+3) = FACI
Y3 * O.¶E2 * Z1 /
KSI (J+4) = FACI * Y1
KSI (J+6) = FACT a Y2 * Z1 / P1
KSI (J+7) = Y3 * KSi (J+6)
C
C
C
FRM,02280
FR 32290
FR:I32330
FR102310
FPA;02320
PFA02330
FR:102340
FRAI'C2350
FRM02360
FEa02370
PI
1 1 0, 1 1 1
J
C
C
C
FRM02210
F- 02220
Fil1102230
FR102240
F.?02250
PdRi2260
FEm023d0
FRMi02390
FRaM2400
P fi
R02410
FRr102420
F M02430
FR0C2440
FS M02450
N2 0 1 C AND N1-1 N2 1 1 0
.
J = NP1l1 * DIFF .
N2 * 8 + 1
KSI (J+2) = X1P(N1I1I)
* X2P(N2t1) * Y3 * OMR3 * 01.2 * Z1 /
*
(0,511 * h3P2)
KSI(J+6) - XlP(Nll) · X2P(N2M1) * Y3 * Zi
FBM02460
FRM102470
FR.,0248C
FP- ,02490
FR.M02530
FPd02510
FRP02520
FR1M02530
FRM02540
FE. O2550
FRM02560
FRM02570
FRM02580
*
FRM02590
(P2 * O1r,F1)
N1-1
*
(j l
+ L3 -
RIR3)
* CMfE2 /
(OMRfl
PIP2 * R3)
FR102600
FR M02610
F.
FERn02620
J =
i1 * NDI1F + I
FR.M02630
KSI (J+5) = XP (Nltil) * X2Y2 * (C1RB1 * O113
FRMC2640
*
- OiElP1 * OnP2 * CME53P3) / (P1P3 * R2)
FP102650
C
FRM02660
C
Nl 1 A TO
N1 N2-2 A
FOE A = 1 0 0 AND
1 0 1
FRZ102670
C
FRM02680
J = N1 * NDIFF + 9
FRS02690
DO 140 I = 1, N2t2
FF.n02730
FACT = X1? (Nl ) * Y2 * (C1E1 * X2 / Z3 - C.RlP1 * OMP2) / B2?1Ffit02710
KSI (J+4) = FACT * X2P(I)
FRMF02720
KSI(J+5) = FACT * X2P(I) ' Y3
FR102730
J = J + 8 ·
Ff02740
140 Ct;TINUE
F? M02750
Ni
O 1
.1
-153C
C
C
fN1
N2-1
1 0 C ANID N1 N2-1
1
FR nC2760
F.R32770
FR H02730
1
= :1 * :DLPFF + N2A.1 * 8 + 1
KSI(J+4) = (X!P(N1l;1) * X2P(42.r1)
/
*
· ((A1P1 - (t*f.1I1
01ii3 * O?:'2)
O*
J
FR1302 790
(P1 * ( i'2 + X3 - F.2h3)))
.- Z1 t 0.12
¥ *
3 ', Rli'3
*
/
(O:O1p
* K3))
KSI (J+7) = X1P (NlH1) * X2P(N2;1I) * Y3 * Z1 $ GOh2 *
*
(a1 + R3 - i1Hs3 - j13p1) / ( C.i1 * P1P2 * 33)
C
C
C
N1 N2 1 1 0
J
= N1
* NDIFF
+ N2
*
C
P TUFi
C
C *$*'.*
±ND OF F,$,s1
C
END
FRa02820
PfiRM283C
FE3029O40
PFMs2850
?E028d60
3FRM02870
(LAST KSI i;LZMLNT)
8 + 1
KSL(J+6) = (h 1 + 3 - hl. 3 - 1,3P1) * 0
C
X2?(N2t1l) * Y3
Z1i / (C?.i.1 4* R3P1
?F.MO2330
FRY02310
.M2
* OAH3
D3!2)
r*
-3-32830
t ][1?(JN1)
4*
FRn302b90
FR0132930
?FRM92910
FRM 02920
\?RK02930
Ffi'32940
FRP32950
FRs02960
-154-
SUBEiUTINE II1FO(ISTAT,
P.
SUmP)
-INFO010
IN0020
c
C--
-I1F30030
-------
C
THIS SUB3:OUTINE C>2:..i'UTES TIlE P£EiFOiLZANCE MEASUBES GIVEN THE
INFO0004
C
PROBA BILITY VECTOQ.
IFO O 50
IH30060
I.IF00070
P.
----------------------------------------------
-
c
COMdON /PARTAS/
(3),
PP(3),
N3AX
t2)
INF00080
INF00090
C
INTEGER N(2),
INFOCo100
1NF0110
ALPHA(3)
c
REAL*16 h, PP
fiEAL*16 P(NSTAT), SUMP
FR..AL*16 EFF, EQI, EQ2, PS2, PS3, PB1, ?PE2,
AQ2, INVENT
*CHFPF(3), AQ1,
IPFOl120
rJTIME(3), DNTIrFE(3),
C·
LOGICAL
BEG
C
C
C
C
C
C
C
C
C
C
NgG
EFF
EQ
EQ2
PS2
PS3
PB1
PB2
=
=
=
=
=
=
=
EFF
EQt
EQ2
PS2
PS3
PB1
PB2
IS
IS
IS
IS
IS
IS
IS
.FALSE.
O.OQ+O
O. OQ+O
O.OQ+O
O.OQ+O
O.OQ0+O
O.OQ+0
0.0+.
THE
THE
THE
THE
TdE
THE
THE
INF00250
·
LINE EFFICIENCY
EXPECTED SIZE OF QUEUE
EXPDECTED SIZE OF QUEUE
PROBABILITY OF 5ACHI.NE
PROBABILITY OF MACHINE
PROBABLITTY OF MACHINE
PROBABILITY OF MACHINE
IFP00130
ItFO0140
IFcOt150
INF00160
INFO0170
11700180
INF00 190
INFO0200
INFO0210
INFO1220
INP00230
INF00 20
1
2
2
3
1
2
STARVED
STAEVED
BLOCKED
BLOCKED
DO 100 I=I,NST
IF (P(I) .LT. Q.OQ+O) NEG = .TRUX.
CALL STOFN (I, , ALPHA)
ALPHA(3) .EQ. 1) EFP = EFP + P(I)
IF (N(2) .GE, 1 .AND.
EQ 1 = EQ1 * N(1) * P(I)
BQ2 = EQ2 + N(2) * P(I)
ALPHA(2) .EQ. 1) PS2 = PS2 + P(I)
.EQ.
0 .AND.
IF (N (1)
IF (N(2) .EQ. 0 .AND. ALPHA(3) .EQ. 1) PS3 = PS3 + P(I)
IPF (N(t) .EQ. NnAX(1) .AND. ALPHA(1) .EQ. 1) PB1 = PBI + P(I)
IF (N{2) .EQ. NnAX(2) .AND. ALPHA(2)
EQ. '1) PB2 = PB2 + P(I)
100
CONTINUE
C
.
DO 110 1=1,3
UPTIEi (I) : 1.OQ+O / PP(I)
1.0Q+0 / E(I)
DNTIME(I)
lC.HZPF (I) = E(I) / (X(I) + PP(I))
F. (I),
DNTIME(I), iCHEFF(I)
WRITE (6,500) I, PP(I), UPTIME (I),
CONTINUE
10
WRITVE(6,540) IMkX
STORAGE CAPACITIES : ',2110,/)
540 FORMAT(1H0,/,.'
iNF00260
INFO0270
IFO00280
IPNc290
I1P00300
INF00310
IN F00320
INF00330
INF00340
I F100 350
INFOO00360
INF00370
INF00380
INF00390
IN10400
INF0010
INF00420
INF00430
IN00440
IN00450
IFO0460
INPFO070
INF0080
INF00490
INF00500
INF00510
INFO00520
INFOO530
INFO0540
INF00550
-155-
AQ1 = EQ1 / QFLOAT(NMAX(1))
AQ2 = EQ2 / QFTLOA.T{(NAX(2))
INVENT = EQ1 + FQ2
C
C
C
C
INF00560
INF00570
INFOO530
I NFO 590
WRITE(6,510) EFF, EQ1, AQ1, EQ2, AQ2, INVENT
INFO0600
WRPTE(6,520) P31, PB2, PS2, PS3
INFOO610
WITE(6,533) NEG,SUdP
INF00620
INFC0630
50C FORMAT(/,' :ACHINE ',I13,' FAILURE PROBABILITY ',F9.6,
*
*'
iEAN UP-TIME
',FlO.3,/,11X,'
EPAIR PROBABILITY
',F9.6, INF00640
*
MEAN
L
DOWN-TINE ',FP10.3,/,11X,'
EFFICIENCY IN ISOLATION ',
INFO0650
INFO0660
*
F6. 3)
510 FORiNAT(/,'
LINE EFFICIENCY ',F10.6,//,
INF00670
*
' EXPECTED STORAGE LEVELS AND FRACTICN OF MAXIMUM STORAGE ',
INFO0680
INFOG690
*
/,'
STORAGE 1 ',F1O.4,5X,PF6.4,
INFO0700
STORAGE 2 ',F13.4,5X,F6.4,/,' TOTAL EXPECTED INVENTORY '
* /,'
INFOO710
* ,F1O.4)
INFO0720
2,F6.4,
520 FORMAT(/,' PROBABILITY GF MACHINE 1 BLCCKED
*
/,'
PROBABILITY OF MACHINE 2 BLOCKED ',F6-4,
INF00730
INF00740
',F6.4,
/,' PFOBABILITY OF NACHINE 2 STARVED
*
INFO0750
* : *
/,*
PROBABILITY OF aACHINE 3 STARVED ',F6.4)
THAT THERE WERE NEGATIVE PROBABILITIES',IFP00760
530 FOi.MAT(/,' IT IS 0,L4,'
INFOO770
- e*
/,'
ORIGINAL SUN OF P WAS',Q10.3,/
INFOO780
BETURIN
INF00790
INF00800
+***t
END OF INFO
INF00810
INF00820
END
-156-
SUBROUTINE INIT (KSI, NSTATE, NOCE, ODSTAT)
C C
---C
-THIS-SUBROUTINE INITIALIZES VAR-ICUS VALUES OF THE PROGRAM AND
C
SETS UP -ODSIAI', THE ARRAY CF CDD STATE-S.
C
.
C
eN' ABE THE 'PADDED' STORAGE LEVELS.
C
THE COMMON BLOCK 'QUAD' CONTAINS THE FACTORS NEEDED TO SOLVE C
THE QUADiA-TIC EQUATICN USED IN OETAINING (Xl,X2,Y1,Y2,Y3)-.
C
I'USOLV' IS USED TO SCLVE FOR (X1,X2,Y1,Y2,Y3).
C--……
C
C
C
C
C
C
C
C
----
IN1OO010
INI00020
N100030
003----------------INI-00040
1-NI00050
-IN100060
INIOO0070
INI-00080
INI00090
I NI00100
--------------------- 10---INI001 10
-
INIO00120
COMMON /P-RA.IS/ R(3), P(3), NMAX(2), MAX(4)
INIQO0130
COMMON /STORfES/ N(4)
IN-I00140
COMMON /FACTO-R/ I1, E2, R3, P1, P2, P3, CMRi, 01M2, OMR3, OmP1,
INI00150
*- 0HP2, OMP3,-OMRIP1, OYR2P2, CMR3P3, R1P2, FiR3, RlP1, R1P2,
IN100160
*
R1P3, R2R3, R2P1, R2P2, R2P3, E3PI,
M3P2,f3P3, P1P2, P1P3, - INI00170
*
P2P3, Nl, N2, NIMI, N2,1, Nl12, N21M2,. NDIFP
INIOOO80
CCMMON-/jUAD/ Al, A2, B1, B2, B3, C1, C2, El, E2, E3
IJNIO0190
CO-MtCN /USOLV/
ALPHA (3) ,BEIA (3) ,GAMMA(3)
INI00200
INI00210
INTEGER ODSTAT(NODD)
INI00220
I-NTEGER NMAX,'MAX, NSTATE, NCED, N
IN100230
INTEGER N1, N2, N1,
N21, N12, N2M2,
22, NDIPPFF
-INI00240
INIO0250
REAL*16 KSI (NSTA'E)
I N1100260
REAL*16 R, P
INI00270.
REAL*16 E1,R2, 2
i3, P1-, P2, P3, OMR1, O-MR2, O0R3,. OP1,
INI00280
*
OflP2, OMF3, O145P1, OMR2P2, CMR3P3, 11R2, R1R3, R1P1, R1P2,
INI00290
-* A-1P3, R2R3, R2P1, R2P2, R2P3, -3P1, E.3P2, R3P3, PIP2, P1P3,
INIO03000
-*
2P3
1NI00310
REAL*16 A1, A2, B1, B2, B3, C-l, C2, El, E2, E3
INI00320
RBEAL*16 ALPHA, BETA, GAMMA
INI00330
.IN100340
THE PEECOMPUTED FACTORS USED IN THE PROGRAM ARE CALCULATED.
INI00350
INI00360
DO 1- I=1, NSTATE
.INIO0370
KSI(I} = 0.OQ+0
1NI00 380
1 CONTINUE
INI00390
.
I NI-00400
MAX(1) = 4
INI00410
MAX(2)
NAX (1)
N=
INI00420
MAX(3} = NMAX(2)
INI00430
TIN00440
MAX(4) = -4
-I NI00450
N(1) = 2
N(4) = 2
.
INIC00460
-IN00470
R1 = R({1)
IN00480
R2 = R(2)
-N1I00490R3 = R(3)INI00500
P1
-P(1)
P2 =
P3-=
N1 =
N2 -
P(2)
P(3)
NMAX(1)
NMAX (2)
-
INIOOS10
INI00520
INI00530
_INI00540
INIoo0550
-157-
Nu11 = N1
N211 = 12
N1M2 = N1
N212 = N2
NDIFF = 8
-
- 1
- I
- 2
- 2
* (N2 + 1)
1 '1I00590
INI)O0630
iI13C610
C
-IN100620
cO1 AiiO2 =
0tRi3 =
O'P1 =
ONP2 =
C.P 3 =
OaA121
o01i:2?2
O;4R3P3
C
c
C
C
1. ):+03 - F.1
1. JQ+O - h2
1.0Q+0 -- 1(3
1.0Q+C - P1
1.0Q+0 - P2
1. 02+0
- .P3
= 1.OQ+0 l(1 - P1
= 1.JQ+0 -h 12 - P2
= 1.OQ+0
3 - P3
-
f15(2
=
5113
R1P1
51P2
P 1.P3
h21;3
ni2P1
P.P2
Ei2i3
h3p1
R3P2
R323
P1P2
P1P3
P2P3
= 1l t
= R1 *
=l
*
= k1l *
= Ef2 *
= R2 *
= R2 *
= n2 *
= h3 *
= n3 *
= K3 *
= P1 *
= P1 *
= P2 *
Fh1
I N00630
INI!0640
II103650
INIC0660
I NIO)670
INIOC680
INI09690
INI03700
INIJ30710
7120
.11i3
* h2
IN103730
F3
P1
P2
P3
h3'
P1
P2
P3
P1
P2
P3
P2.
P3
P3
.
-
-
.
31 = OP1 * OP3
3B2 = G5i1iP1 * 01]3P3
B3 = OMR1 * 01O3
INI00890
IN1O0900
I100910
INI0092C
INI00930
INI00940
1TIO0950
-
.1100960
C
IIO0970
INI00980
1NIOG990
IY 01000
IN110101
Cl-= 05R1
C2 = OSP3 * OMR1P1
C
El = 011E2
E2 = 0MB2P2 /
E3 = 01P2
OP2
INlIOt020
INI01030
INI01040
1IO01050
INI01060
C
DO 5 I=1,3
ALPIA (I) = 1.OQ+O - P(I)
BETA(1) = (1.OQ+O - R(I) - P(I))
cGA, AA(I) = 1.00+0 - a(I)
5
CcITINtUE
IN100740
IN10750
IN10760
INI0770
INIOC780
INI33790
IN100800
IRI00810
IN100820
NIIOC830
INI30840
INIM03850
I1100860
1I130870
-INI00880
THE FACTORS FOR QUAD ARE CALCULATED.
A1 = 10A3
A2 = OIP1 * O0R3P3
C
C
1NI930560
I I30570
IiI,3580
/
( 1.OQ+O
-
P(I))
iN:01070
lIt01080
I I01090
1I01100
-158-
C
C
C
C
C
C
C
C
C
C
C
GENEFATE THE INDIC$3 OF THIE ODD STATES.
TH±ERE AbE 4 *- (I1 + N2) - 10 ODD STATES.
FIRST IiL 6 CORNER SIATES
0 1 0 t 1
CDSTAT(1) = 12
1 1 1
1 .
ODSTAT({2) = NDIXF. + 16
1 N2-1
1 1
1
ODSTAT(3) = NDIFF + 8 * N2,11 + 8
N 1-1 N42-1 11 1.
ODSTA (4) = N1' 1 * INDFF + N211 * 8 + 8 .INI01230
. N-1
N2 1 11 0
ODSTAT(5) = N1M1 * NDIFF + N2 * 6 + 7
N1 0 1 0 1i
ODSTAT(6) = N1
NDIPF + 6
C
CC
C
C
C
1 N2 0 1 C
1 N2 1 1 0
2 0 0 3 1
2 0 1 0 1'
TO
N1-2 N2 0 0 1
TO
1-2 N2 1 0 1
TO
N0 0 0 1
TO N1-1
0
1 0 1
.
111101270
INTO 1280
1NI01290
INI01300
IN101310
INI01330
1
INI01340
11o101350
32 = 2 * NDIFF
INI01360
+ 2
ODSTAT(ICNT+1)
= J1
IN131370
INI01380
+ #4
ODSTAT(ICNT+2) = J2
ODSSAT(ICNT+3) = J2 +
ICNT; = ICNT + 14
31 - J1 + NDIFF
J2 = J2 + NDIFF
CONTINUE
10
INI01390
INI01400
IN101410
INIO1420
4
INI01430
.
0 2 0 I 0 TO
0 2 0 1 1 TO
Ni 1 I 00
TO
N11 1 1 0 1
0 N-1
01 0
0 N2- 0 1 1 1
N1 N2-2 1 0 0
N1 10N2112 10 1
~~C
J1 = 19
J2 = N1 * NDIFF + 13
DO 20 I=,
N2M2
ODSTAT(ICNT) = J1
ODSTAT(ICNT+1) = J1 + 1
ODSTAT (ICNT+2) = J2
ODSTAT(ICNT+3) = J2 + 1
ICNT = ICNT + 4
J1 = 1 + 8
.
J2 = J2
+ 8
CONTINUE
EETURN
C
C
INI01260
ICNT = 7
J1 = NDIFF + N2 * 8 + 3
DO 10 I=1,N112
ODSTAT(ICNT) = J1
20
INIC1240
INI01250
-INI1320
-
C
C
C
C
C
INI01110
I NI01120
INI01 130
IN101140
INI01150
INI01160
NIO 1170
I I0180
IN
INI01190
INI01200
IINI01210
INI01220
*****
END OF INIT
C
END
IN101440
IN1101450
IN101460
IN01470
INI01480
INIO1490
INI01500
1u~-INI01510
INI01520
IN101530
INIO01540
INIO155C
IN101560
INI01570
INI01560
INI01590
IN101600
INI01610
IN101620
INI01630
1NI01640
IN101650
INIO
1660
INI01670
.- 159-
SUBKOUfINE INTKSI(I, DIMU, DI¶KSI, KSI, U. NFODD)
c
--------------------C--THIS SUBROUTINE COMPUT'£S TH7 ELEHENTS OF THE KSI VECTGE WHICH
C
C
COERESPOND TO THE INTERNAL STATES.
C
C
THE ELEMENT FOR STATE (N1,N2,A1,A2,A3) HAS THE FORK:
3**A3
*
Y1**A1 * Y2**&2 *
C
X1i*N1 * X2**N2
C
THE INTEFfNAL STATES RUN FRON N=2 TO nMAX-2 FOR BOTH N1 AND N2
C
C
C
FOP. CO0;PUTATICNAL EFFICIENCY COhMON PEODUCTS ARE PECO'!PUTED.
….
......
C-----C'
COHtON /PARA!¶S/ E(3), P(3), NMAX-(2)
C
INTEGER DI!U, DINKSI
INTEGER NtMAX, I, NFODD, J, N1, N2, NDIFF, IA1, IA2, 113
INTEGER N1N2,N292.
c
C
Xl
12
I
Y2
13
1)
U(I, 2)
U:(l, 3)
4
(I. q)
U(1, 5)
= U(IX,
I AND X2
SET UP THE PRODUCT ARRAYS OP X
FACT
XI
2
NMA(1)312
IF (N31X(1) .LE. 3 OR.0 NHAX(2)
DO 10 J = 1, 1182
iP2(J) = FACT
FACT = FACT A X1
10
CONTINUE
FACT - 12.
1232 = WMAX(2) - 2
DO 20 J= 1, 1252
X2P(J) = FACT
FACT = FACT * 12
20
CON TINUOE
C
YIP(1) = 1ITg.0
Y2P(1) = 1.OQ*0
Y3P(I) = 1.0Q+0
1IP(2) = 1
72
72?(2)
713
!3P(2)
8 * (BHA1(2)
NDIFF
1)
I
INT00210
INT00220
INT00230
IT00240
INT00250
INT00260
INT00270
INT00280
IMT00290
14T30310
INT00310
INT00320
INT00330
117··INT00340'
c
C
INT00130
INT00 140
INTO0150
INT00160
INT00170
1-NT00180
INT0O190
-INT0o200
REAL* 16 KSI(DIIIKSI), UI(NFODD, .5)
REAL*16 e, P
REAL*16 X1P(100), X2P(100), 11P(2). Y2P(2), T3P(2),
Xi, X12,
1, ,2,
Y3, FACT'
C
C
INTOO10
INT00020
-INTOO030
INT3O040
INT03050
INT00060
INT00070
INTO0080.
INT00090
INT0O100
INT00110
II{T00120
INT00350
.LE.
3) RETURN
INT00360
INT00370
I11T00380
INT00390
INT00400
INT00410
IT00420
I1T00430
IT00440
INT00450
1NT00460
INT00170
13700480
I00490
INT00500
INT00510
INT00520
I1700530
INT00540
I NTO0 550
-160C
C
CO:iPUTS THE INTEENAL STATES
.
DO 70 N1 = 2,NIM2
DO 60 N2 = 2,N2M2
J = Nl * NDIFF + N2 * 8 + I
DO 50 IAl = 1,2
DO 40 IA2 = 1,2
DO 30 IA3 = 1,2
C
*
KSI(J) = XlP(N1) * X2P2N2) * YlP(IA1) * Y2P(IA2)
Y3P (IA 3)
J =J
+ 1
C
30
CONTINUE
CGONTINUE
40
50
CONTINUE
60
CONTINUE
70 CONTINUE
BETUI.N
C
C
C
,***#END OF INTKSI
END
'
*
.
INTO0560
INT00570
INT00580
INT00590
INT00600.
INT00610
INT00620
INT00630
INT00640
INT00650'
INT00660
INS00670
INT00680
I NT00690
I NT0700INT0710 INT00720
INT00730
INT00740
INT00750
INT00760
INT00770
INT00780
-161-
FUNCTIGN ID(IX)
C
C
C
-
------------------------------THIS SUBRDUJTINE IAKZS INrO ACCOUNT THE CONDITION
IN WHIi:i A DONSTPEEA1 NACHINE IS BLOCKED.
C
C
CO-MON /:tS/ I4(2)
COi.1o
/PAFiA$1S/
h '3),
.
P :3), NP t2)
C
INTEGER ID,I,
.C
IN,NP
.
REAL*16 h,P
C
IF :IX .EQ. 2) 30 TO 1
IF 'IN:IX * 1) .NE. 1 + NP:IX + 1)) GO TO 1
2 ID
O
0
RETURN
1 IF
IN IX) .EQ.
1
)
0 TO- 2
ID = 1
BETURE
C
.
.
C ***** gND OF ID
CEND
162
---
BI'O1603
2'I01610
IO 0
Pi131633
PM'I01640
P/1t01650
PM1i01660
M101670
P1I)1680
PMI 01690
P31 1700
PII 0171 0
P£1I01720
PM IO1730
PMI01740
PM101759
PM101760
PSIU 1770
PH01780
PHIO 1790
?n 101800
PMI01810
PMI01820
P1I01 830
PSIO 1840
-162-
INTEGER FUNCTION ITRANS(NMACH)
C
c --------------------------------------------------------------C
THIS FUNCTION RETUENS 1 IF THF MACHINE IS OPERATING.
C
IT RETURNS 0 IF THE MACHINE IS FORCED DOWN.
C
C
'MAX'IS THE MAXIMUM CAPACITIES OF TtHE STORAGE ARRAT PADDED WITH
C
FICTITIOUS STORAGES I AND K+1. THE FORMER IS NON-ESPTY AND.
C
THE LATTER NON-FULL..
C
C
'N' IS THE CURRENT LEVELS IN THESE 'PADD2D$ STORAGES.
C---------------------------------------------------------------C
.
.
COMMON /PARAMS/ R(3), P(3), NMAX(2), NAX(4)
CCfcN /STORES/ N(4)
c
INTEGER NIMACH
iNTEGER MAX, NMAX
INTEGER N
C
REAL*16 R, P
C
ITRANS
1
IF (N(NMACH) .EQ. 0 .OR. N(NMACH + 1) .EQ. MAX(NMACK + 1))
ITRANS = 0
RETUEN
C
C ***** END OF ITRANS
c
END
ATEfi00750
ATR00760
ATR00770
ATR00780
ATROC790
ATR00800
ATROO810
ATR00820
ATR00830
aTEo008o
ATE00850
ATROC860
ATRC0870
ATROC880
ATI00890
ATROC900
ATRO09 1O
ATROO920
ATR0930
ATR00940
ATRO0950
ATROO960
ATRO0970
ATR00980
ATR00990
ATRO1000
AT~.10101
ATR01020
ATR01030
ATR1040
-163-
FJnr IO4 I[I fIX)
.
C
..
C
C
TiHIS FUNTIOII TAKES INTO ACCOUNT TUE CON.DTI3rD
IN WHICH A. 0PSTREAA MACHiNbE IS STAEVED.
_
----------------------------------------------C----------------------
~C~~~
COa:O1N /$NS/ I'N2)
COHMON /PARAIS/ R(3), P(3), NP(2)
C
C
-PHIO
INTE(;ER IU,IN,NP,[X
.REIL*16 R,P
IF (IX .EQ. 1) GO TO 1
1
IF (IN IX - 1) ,NE.
2 IU = 0
RETURN
1 + NP(IX))
I IF (INfIX) .EQ.
IU = 1
RETURN
-
C ***** END OF 1c
END
) GO TO 1
3)
TO 2
-
Pn1101330
?Ml 1340
19135)
Pn101360
Pn1111373
I1383
P~~~
.9101390
P ilO1 400
PM 3 1413
I422
PHI01430
P·1i)1443
- PMI01450
PNaI3463
PMI01470
PZ101480
paRII490
P. I01500
PIu01510
PMI'01520
. P3IO1530
PO101540
P1101550
PRM1560
-164-
SUBROUZIN. LIMKSI(KSI,NSTATE,JKSI)
LIM00Q10
C
LIM00020
C----------------------------------------------------------------------LIMO30
C
COMPUTES ANALYTICALLY TIE LIMITING KSI VECTORS
LIM00040
C
FOR LIMITS AS uX, X2 OR BOTH GO TO ZERO OR INFINITY
LIMOO)50
C---------------------------------------------------------------------LIM 060
C
LIM00070
COMMON /P ARAMS/E (3) ,P(3) ,NSTCR (2)
LIM00080
COMMCN/FACTOR/R1,R2,E3,P1,P2,P3,GMR1,OMR2,OMR3,OMiP1,OMP2,OMP3,
LIM00090
*
Q
iOHMP
1,R,0P2L 2, OMP3R3, filR2, R R3,R1P1,.l P2, R1P3, R2i3, R2P1,
LIMO 100
*
R2P2,R2P3,R3P1,R3P2, .3P3,P 12,PIP3,P2P3,N1,N2,N11,N21,
LIM00110
*
N12,N22,NRIFF
LIMOO 120
C
LIM00130
REAL*16 KSI (NSTATE),R,P,R1,rE2,R3,Pl,P2,P3
LIMO0140
REAL*16 OMR 1 ,OtR2,
OIR3,OMP1, CMP2, CMP3,OiFP1R1,OMP2R2,OP.3R3
LIM00150
REAL*16 R1PI,R1P2,R 1P3,R2P1,R2ZP2,R2P3,R3P, R3P2,R3P3
LIMo0160
R3AL*t6 R1R2,R 13,R2R3,P1P2, P1P3,P2P3
LIM00 170
REAL*16 X1,X2,Y1,Y2,Y3,Z 1,Z2,Z3,Q1,2,Q3
LIM0180
C
LI00190
INTEGER NSTOR,N 11, N 12,21,N22,N,NSTATE,NDIFF,N1,N2
LIM00200
INTEGER IN, IOUT, JKSI
LIO00 210
C
LIH00220
GO TO (100,200,300,400,500,600,700,800,900,1000, 1100, 1200) ,JKSI
LIM00230
C
LIMO240O
100 CONTINUE
LIM00250
C
-------------LIM002.--60
C
LIMITING CASE NUMBER 1
LIM00270
C
(LOWER LEFT CORNER - Xl AND X2 GO TO ZERO)
LIM00280
C
(CURVE: +-+ -)
LIM00290
C
(NORMALIZING: X1 * X2 * Y).
LIMOO00300
C---------------------LI---MO31-------------------------------------L003
Q2=ONR3*(O'-R3-ONMP1I*MP2*0MP3R.3)/(OPl*OMP3R3*(OMR3*0MR2LIM00320
*~OM
Pl*OP3Ph3*CNP2R2))
LIMC0 330
LI K00340
Z2=OMR3/ (Q2*OMP1*OM P3R3)
TY2= (Z2-O1i2)/P (2)
LIM00350
Y3=-OOMR3/P(3)I
LIM00360
c…….
.......---------.-------------LIM00370
CALL ZEROC (KSI,NSTIAE)
LIM00380
KSI (4) =OMRI* (R(1) +R (3)-R1R3-R 1P3) *Y2/ R (1)*RIP3)
LIM00390
INDEX=NOFST2 (0 ,10, 1,0)
LIM00400
KSI (INDEX)-OMR1*Y2/R (1)
LIM00410
KSI (INDEX+1)= (R (1) +i (3)-E13) *Y2/RlP3
LIM00420
INDEX=NOfST2(1,0,0,0,1)
LI00430
KSI (INDX)=R3P2*P (1)*Y2/(RP3*(R (1)+R(2) -R R2))
LIM00440
KSI (NDEX+6)=
(1) +E (3)-R 13-R1P3) *Y2/R 1P3
LIMd00450
INDEX=NOFST2 (1,1,1,1,0)
LIM00460
KSI(INDEX)=Y2
LIM00470
KSI (INDEX+1)=Y2*Y3
LIM00480
INDEX=NOFST2 (0,2,0,1,0)
LIM00490
KSI (I NDEX) =(OaR3*CMPf2/OMP3R3-O IP1*OP2R2)/R 1P2
LIM00500
KSI (INDEX+1) =KSI (INDEX)*Y3
LIM00510
GO TO 9999
LIM00520
200 CONTINUE
LIM00530
C
LIM00540
C
----------------------------------------------------------LI
550
-165C
L1,')0560
LIMITING CASE 113Ji.3ER 2
C
(LO;{ER LLFT CORNItE, - Xl ANID X2 GO TO ZERO)
LIM00570
C
(CURVE: - + +)
LI;OO0580
C
(NORIALIZIUIG: X1 * X2)
C ----------------------------------Y1--R (1)/O1P1
L2 =O
P1/ (O
LIMOC590
----------------------------------- LIM00600
LI'lOG610
P1. l*ON4f3)
LI OO 620
Y 2= (Z2-0.R2)/P (2)
LI100630
L
C---------------------------------------------------------------------- LIOO6406O
CALL ZERO(KSI,NSTATE}
KSI (4)=OCR1* (t
(1)+R (3)-R 1R3-R 1P3) *Y1Y2/(R( 1)*R1P3)
INDEX=NOCFST2 (0, 1,0,1,0)
KSI (INDEX)=OiR 1*Y1*Y2/R (1)
K S (INDEX+1 )=
(1 )+R (3) -R 1R3) *Y1 *Y2/R1P3
INDEX=NOFST2 (1,0,0,0,1)
KSI(INDEX)=( F3P1P(2) Y1I*Y2/E (1)+GOP3-O1P3
3*CO.R1O
1
*
(P (3) * (hi( 1) + (2) -E 1R2))
KSl (I NDEX+6) = (R (1) +E (3)-R1R3-R 1 P3) *Y1*Y2/1 P3
KSI (INDEX+7) =1 .QO
KSI (INDEX+ 1 3) =OaP3*Y 2/P (3)
KSI (INDX +1 3) =Y 1*Y2
KSI (INDEX+14) =0O,tE2*0MR3*Y1/P2P3
INDEX=NOFST2(0,2,0,1,0)
KSI (I NDEX) =-OMP1 AOAP2 R2*Y 1/B 1P2
INDEX=NOFST2 (2,0
,0,0,1)
KSI (I ND.K) =M0E3*01aP1*Y2/ (R2P3*OMP 1R 1)
KSI (INDEX+4) =KSI (INDEX) *r 1
GO TO 9999
300 CONTINUE
.
C
C -------------------------------------------C
LIMITING CASE NUMBER 3
C
(LEFT EDGE AND CORNERS - X1
C
(CURVE: 4 - +)
C
(NORMALIZING: XI * Y1)
C
------------------GOES TO ZERO)
.L------------------------------------
Q3=0 1MR2* (M01R2-O!i P 1*0 P3*O P2E2)/ (OMP 1 *.
t*
GO1dP1*OMP2R2*OMP3R 3))
Z3 =OMP2/(Q3 *OlP1*O3 P2R2)
Y3= (Z3-O1R3)/P (3)
Y2=-0 MR2/P (2)
X2=1.QO/Q3
C
.E2)/
LIM00650
L00660
LI.900670
LIhOC680
LI M30690
LIM0700 0
LI.NOO710
LI i00720
LI100730
LIM007'40
LI100750
LIMf100760
LIM00770
LIM00780
LIMOO 790
LIHMO0800
LIMOo810
LIM00320
LI-00830
LI00C840
LI100850
-- L
00860
LIM00870
LI M00880
LIM00890
LIM00900
LI0910
P2a2* (O
2*onM3-
---------------------------------------
CALL ZERO(KSI,NSTATE)
KSI(4) =OR1*R((1)+R (3)-R1R3-RP3) *X2*Y2/(R(1) *1P3)
INDEX=NOFST2 (0,, 1 1,0)
KSI (INDEX) =OM 1*X2*Y2/R (1)
KSI (INDEX+1l)= (R (1) +R (3) - 1R3) *X2*Y2/P3
INDEX=NOFST2 {1,0,0, 0,1 )
KSI (INDEX) =R3P 1 *P (2) *X 2Y 2/ (lP3* (R(1) +1F(2) -B1E2))
KSI (I NDEX +6) = (R ( 1) +R (3) -E 1F 3-F 1 P3) *X2*Y2/R1 P3
IN DlX=NOFST2 ( , 1 1, 1,C)
KSI (INDEX) =X2*12
KS I (INDEX+ 1)=X2* (Oai 2*Z3 +P2P3*Y2*Y3) /P2P3
GO TO 9999
LI M00920
LIM00930
LI M0940
LIM00950
LI M00960
LIi100970
LI00980
LIM00990
LIMO1000
LIEO1 010
LIM 1020
LI01 030
LIMO1040O
LIMO1050
LI MO 1060
LIM01070
LIM01080
LIMO1090
LIMO1100
-166LIMO 1 110
LIM11120
LIM01130
LIMO 140o
LIIO1150
400 CONTINUE
C
C------------------------------C
L;'!'TI-NG CASE NUMBER 4
(LEFT EDGE A'D CGRNERS - X1 GOES TO ZERO)
C
(CURVE:
C
-
LIM01160
+ +)
LIMOl1170
LIO110
LINt01190
LIa01200
LIM1210
(NORMALIZING: Xl)
C
C-----------------------------------------------------------------Z3=OMP1/ (OIP1RI1COMR2)
'Z3 -CM P3R3) /(Z3* (Z3-OME 3) )
Q3 = (C3i3
X2=1. QO/Q3
LIMO1220
LIM0 1230
Y1=-H (1)/Cp!P
Y3= (Z3-OiIR3)/P (3)
--
C----------CALL Z EO (KSI, NSTATE)
IND&X=NOFST2(1,0,,0,1)
KSI {(INDEX) = (Ci.IP3-OiIP3R3*-OMi1*OGME2) *X2/(P (3) ({R (1) +R (2)-R1R2) )
INDEX=NOFST2 (1,1,0,0,0)
KSI (INDEX) X2
KSI (I NDEX +1) X2* Y3
KSI (INDEX+7)-X2*Y1*OMR2*Z3/P2P3
lNDEX=NOFST2(0,2,0,.1,0)
DO 401 N=2,N21
2E2)/R1P2
KSI(INDEX) =X2 **(N-1)* Y1* (OMR2*X2/Z3-CM P1 **OP
KSI ({INDEX+)=KSI(INDEX) *Y3
401
INDZX=INDEX+8
INDEX= NOFST2 (1,2,0,0,0)
DO 402 N=2,N22
KSI (INDEX)=X2**N
KSI (INDEX+1)=KSI (INDEX)*Y3
KSI (INDEX+6) =KSI (IN DEX) *Y 1eOR2/P (2)
KSI(INDEX+7)=KSI (INDEX)*Y1 *Y3*OMR2/P(2)
INDEX=INDEX+8
402
INDEX=NOFST2(0,NSTOR (2).0,1,0
KSI (INDEX)=X2 **N21* (GMP2-OMP2R2*OR1*OM 3)/(P (2)
{(B(1) +R(3) -i1E3))
INDEX=NOFST2(1,N21G.0,00)
KSI (INDEX)=X2**N21
KSI(INDEX+3)=KSI(INDEX) *Y3*OMR2/P (2)
KSI(INDEX+6)-=KSI(INDEX) *Y 1 *CM2/P(2)
KSI (INDEX+7)=KSI (INDE;X) *Y1 *Y3*O MR2/P(2)
INDEX=NOFST2 (1,NSTOR (2) 0, 1,0)
KSI(INDEX)=X2**N21*Y3*OMRE2/(R3P2*(CnR1+P (1) *Y1))
KSi (INDEX+4)=KSI (INDEX) *Y 1
GO TO 9999
500 CONTINUE
C
LIM01240
LIM01250
LI101260
LIM01270
LIM 01280
LIM01290
LIMO1390
LIM01310
LI01320
LIMC1330
LII.013 40
LIM01350
L1031360
LIM0 1370
LIMO1380
LIMO1390
LIM01400
LIMO1410
LIM01420
LIM01430
LIMO 144.0
LI n01450
LIO01460
LIH01470
LI M01480
LIt01490
LI101500
LIMO1510
LIM01520
LIM01530
LI01540O
LIM01550
LIM01560
LIMO1570
L…I…01580
C…---
C
C
LIMITING CASE NUSBER 5
(LOWER EDGE AND CORNERS - X2 GOES TO ZERO)
LI101590
LIM01600
C
(CURVE: + + -)
LIHO1610
C
C
LIM01620
(NORMALIZING: X2 * Y2)
--------------LIO1630
…………L1M0_________---LIO1640
Q1=CME3* (On R3-OMPP2*OMP1 *OMP3Eb3) / (CP2*OrP3R3* (OHR1 *ONR3LI101650
.*
01MP2*03P3R3*0PIP1R1))
.-167Z 1-=0fR3/(Q1 Oi'P2*Of1P3P3)
LItO 1660
X1=Q1
LIt01670
Y1= (Z1- R1)/P (1)
LI M01680
Y 3= - OBR3/P (3)
LI 10 1690
C -------------------------------------LI1
700
CALL ZERO(KSI,NSTATE)
L I 13 17.10
KSI (4)=OMR 1- (R ( 1) +R (3)- R31
R I1P3) *X 1*Y1/{H (1)*R1P3)
LIM01720
IND)X= NOFST2 (0,0,1
1,0)
LIM01730
KSI (I~DEX)=0o il 1X 1*Y 1/R (1)
.LIM01740
KSI (INDEX +1)= (R (1)+R (3)-H 1F3) *X I* Y 1/R1P3
LIM01750
INDEX=NOFPT2 (1,0,0,0,1)
LIIM01760
KS I (INDLX) =.Pj21 tP (2)-'X1 *Y 1/ (R1P3* (E( 1)+P (2)-a 1R2))
LIM31770
KSI (INDLX +6) = (R (1)+R 3) -R 1R3 -Rl P3 )*X1 *Y1/81 P3
LIM0 1780
IND-EX=NOFST2 (1,1,1,1,0) .
LIN01790
K SI (INDEX) = X 1*Y
LI 01800
KSI (INDEX+1)=KSI (INDEX) *Y3
LIM01810
INDEX=NOFST2 (NSTCa (1),1,1, o, 0)
LIa01820
KSI (INDE X)=X 1 **N 1 1 (0.A 1OR 3/0 MP33-OMP2*OMP1 1)/R2P1
LIM0 1830
KSI (I NDEX + 1) =KSI (INDEX) *Y3
LIA01840
INDEX=NOFST2 ({NSTOR(1) ,0,1,0,1)
LIM01850
K S I (I NDBX) =X 1**N 1 1* (O MR1 *OMP3-on P 1 1 *01 P 3R3-OMp2)
LIfM01860
*
/ (P(1)*R2P3)
-LI0 1870
GO TO 9999
LI0i1880
600 CCONTINUE
LIO 1890
C
LIMO1 900
C ------------------------------------------------------------- LI01910
C
LIMITING CASE NUMBER 6
LIA01920
C
(LOWER EDGE AND CORiNERS - X2 GOES TO ZERO)
LIM01930
C
(CURVE: * - +)
LIM019&40
C
(NOP MALIZING: X2)
LIM01950
C ----------------------------LIMO 1960
Z 1 =OP2/ (CGMP2R2*0C! 3)
LIMO 1970
Q 1= (O a71 +Z1 -Ot P1 P1 )/ (Z1 (Z1-OMF 1))
LIL 01980
Xl=Q
.
LIM01990
Y1= (Z1-OMBl)/P(1)
LIM02000
Y 2=- R (2) /OP2
-LIM020 10
C--------------------------------------------------------------------LI02020
CALL ZERO(KSI,NSTATE)
LIM02030
KSI (4) =OM* 1 ( (l1)
+R (3) - 1R3-B 1P3) *X 1Y1*Y2/ ( (1) *R 1P3)
LIM02040
INDEX=NOFST2(0,1,0,l1,0)
LI MO 2050
KSI (INDEX)=CMh i*X 1*Y¥1 *Y2/B (1)
LIM02060
KSI (I NDEX +1) = (( 1 ) +R {3) -Rl 13) *X 1 * I*Y2/ 1 P3
LIA 02070
IND&X=NOFST2 (1,00,0,0,1)
LIM02080
KSI (INDEX)=X 1 *(E3PI*P (2) *1 *Y2/R (1) .+OMP3-0MP3R
3*OlR1*OMa2)
LI 02090
*
/ (P (3) * (R (1) +e (2) -fi R2))
LIz02100
KSI (I(NDEX+6) = R (1) +R (3) - R1 R3-1P3)*X1*Y1* 2/B1P3
LI102110
KSI(INDEX+7)=Xl
LIM02120
KSI (INi)EX+10) =X1*Y2l2*O.3/P(3)
LIM02130
KSI (INDEX+13) =X1*rl*Y2
LI02140
K I (INDEX+14)=X1*Y 1*OM 2*0MR3/P2P3
LIM02150
INDEX=NOFST2 (0,2,0, 1,0)
LIM02160
LIr02170
KSI (INDEX)=-X 1 *Y1*GP 1P*oP2R2/l P2
INDgX=NOFST2 (2,0,0,0, 1)
LIM021 80
DO 601 N=2,N1ll
LIM02190
KS I (INDEX) =X1** (N- 1) *Y2* (0MR3/Z 1-X 1t*01P2*0oP3R3) /22P3
LI02200
-168-
LIN 02210
KSI (INDEX+4}) =XSI (INDEX) *X 1
LIM02220
{(STOR (2)+1})
I`IND.X=INDZXI8t
I.lI02230
INDX=NOFST2 (2,1,C00,3)
L6i-J2240
DO 602 N=2,N12
LIl02250
KSI (INDX)=X1**N.
LI M02260
-- 2*O0R3/P ( 3)
= +3) X 1 }:l3
KS-I{I NDEX
LIM02270
KSI (INDXX+Q) =X1**I*Y1
LI M0228C
KSI (IEDEX +7) =KSI (INDEX+3) tY 1
LIM02290
INDRX=INDEX+8 ({NSTOR(2)+1)
602
LIM 02300
KSI (INDEX)=X1*+N 1
LIM02310
KSI (IN9EX+4)=KSI (INDEX) *Y1
LIiIC2320
KSI (INDBEL+7) =KSI (INDEX) *Y2-Z 1*0OaR3/P1P3
LIM 02330
I NDEX=.OFST2 (NSTOR (1) ,1,1,0,0)
LIMO2 340
KSI {INDSX) =-X1 I**N11*Y2OP2*CAP 1a 1/R2P1
LI n02350
INDI=NIOFST2 (NSTOR (1), 0, 1,0, 1)
LIM02360
KSI (INDEX)=Il1*N 11*Y2*(CO1R1 *GMR3-OP1lRI *OM3R3*0nP2)
LIMf02370
/(?(1)*P2P3)
/{
*
LII02380
GO
'O9999
LI 2 390
700 CONTIhUELI02400
.
C
-LIO210
C ---------------Ll,02420
LIIITING CAS3 NUBER 1
C
LIM02.430
(UPPES EDGE AND COENERS - X2 GOES TO INFINITY)
C'
LIM02440
(CURVE: * - +)
C
LI02450
(NCORALIZING: X2**N21 * Y3)
C
C26-----------------------------------------------------------------------LI2460
LIM02470
/
P3*ON P2R2*(OMR11*O 1R2Ql=OnfR2* (OaP2-C1.P3*OaP1l*O P2 R2)/O
LIM02480
0CMP34OftP2 22CffP 1R 1))
*
LIMO 2490
Z l=OMR2/(Ql*GlP3*CMP2R2)
LIM02500
Xl=Q .
LI 02510
Y1=(Z1-oaR1l)/P(1)
LIhO 2520
Y2=-0132/P( 2)
C---------------- ----------------------------------------------------- LI02530
LIM02540
CALL ZERO(KS.I,NSTATE)
LIM02550
INDEX=NOFST2(N11,N21,0, ,1)
LIB02560
KSI (INDEX.)=X1**11*Z10*N'E2/(P (2)*OMRI)
LIM02570
KSI (INDEX+4=) X 1**N1 1*Y2*Z 1/P(1)
LIM02580
}, 0, 1,0)'
INDEX=NOFST2 ({N1,NSTO (2)
LIM02590
KSI (INDEX)=X1**N 1 1Z1*OMR2*OMlR3/(OMR1*R3P2)
LIM02600
KSI (INDE +4)=X1**N11*Zl* (R(1) +R(3)-R1R3)*0OR2/(OMR1*P (1)*R3P2)
LIMOQ2610
INDEX=NOFST2 {NSTOR (1) N21,1,0,0)
LI ~102620
KSI (INDEX)=Xt**N11*Z I*OR 2*R1P3/ (P3Pl*OiRl# (B (2)+R (3)-R 23))
LIM02630
(1) +R 3})-R 1 3-R3 Pl)*O1ER2/C(OtR1*P (1)*
KSI (INDEX+3)=Xl**+I11*Z1* (R
LIN02640
B3P2))
.*
LI M02650
KSI (INDEX+1 0)-KSI (INDEX+3)*O MR3/R (3)
LIM02660
GO TO 9999
LIn02670
800 CONTINUE
LI02680
C
6--------------C
LI02700
C
LISITINGCASE NUABER 8
LIa02710
(UPPER EDGE AND CORNERS - X2 GOES TO INFINITY)
C
LIM02720
(CURVE: * + -)
C
LIM02730
(NORMALIZING: X2**N21)
C
C2740-------------------------------------LIM02750
Z 1=OP3/ (OfR2*OP3R3).
601
-169-
Q 1= (o iP1.Z 1-oP1R1) /(Z t1 (ZI-ORN 1))
LIM02760
X 1=Q1
LIMO 2770
Y 1= (Z1-O0R 1/ P
)
LI)2!780
Y3=-F (3)/O P3
LI02790
C-.. .------------LI02800
CALL ZERO(KSI,NSTATE) .
LI'02810
NiDEI=CN'OST2
(1,N21,0,0,0)
Lr02820
KSI (IND)EX) =X1
LIM02830
K SI (INDEX +3) =X1*Y3.*n0R2/P (2)
LIM 02840
KSI (INDZJ;X+6)= X1 YICO*AR2/P (2)
LIM32850
KSI (INDEX +7) =X 1 *Y1*Y3*OMR 2/P (2)
LI '12860
I:iDEX=NOFST2 (2,1421 ,0,0,0)
LIM 02870
DO 801 N=2,N12
LI1102880
KSI (INDE) =X**N
LI )2890
KSI (I;i4DiX+3) =KSI (INDEX)*Y3*OMR2/P (2)
LI?102930
KSI (IND-ZX +4)=KSI (INDE;X) *t1
LIa02910
KSI (IND.:X+7) =KSI (INi)EX) *Y 1*Y 3*0 NP2/P(2)
LIM02920
801
IN DeX=INDEi)-X +8* (NST OR (2)+ 1)
LI 102930
INDEX=NOFST2 (0,14STOR (2) ,0, 1,0)
LIM02940
KSI (INDEX)=X!* (OMP2-C[P2 52* Ofiit*OMi3)/(P (2) '(E (1)+R (3)-i 1R3)
LI!M02950
INDEX=NOFST2 (1,NSTG (2) ,0, 1,0)
LIM02960
DO 802 N=1,N12..
LIM02970
KSI (INDEX) =X1 **N*Y3* (OnR2/Z1-X1 0 P2 2*0MP3)/R3P2
LIM02980.
KI3 (INDEX+4)=KSI (INDEX) Y 1
LIMO2990
802
I N DEX =IN DEX +8* (NSOR (2)+ 1)
LIM 03000
INDEX=NOFST2 (N11,N21, ,0, O0)
LIN03010
KSI(INDtX)=X1**N11
LIM03020
KSI (I NDEX +3) = KSI (INDEX) *Y3 *Z 1*01i2/(OCE'l *P (2))
LI O3030
KSI (INDEX+4) =KSI (INDEX) * 1
LIM03 0040
INDEiX=NOFST2 (N11,NSTOR (2),0, 10)
LIM03050
K SI (INDEX) = X1**N 1 1*Y3*Z 1 OMR2*0nR 3/ (OMtR 1 *3 P2)
LIN03060
KSI (INDEX+4) =Xl1**N11 *Y3*Zl* (R (1)+R (3)- 1 3) *OMR2/(OM1R*P(1)*R3P2) LIli03070
IN DEX=NOFST2 (NSTOR (1), N21, 1,00O)
LIMO03080
KSI (INDEX)=X1**N11l*(OiP1I-OCR3*OtIPlRl*OaR2+Z t*O1R2*P1P3*Y3/
LIM03090
*
.
(P 3) *OX1))/(P (1)*(R (2) +R(3)-R2 3))
LIM03100
KSI I NDEX+3) =X 1**N11*Y3tZ3 1* ( (1) +R (3)-PR1e3- R3P1) *CmR 2/(OR8 l*p (1)* LIM03110
*
R3P2)
LI M03120
KSI(INDEX+10) =X**N 11*Y3Z1 *R(1)+R(3) -E 13R3P1)*on 2*0DR3/
LIM03130
(*
O
IR1*R3P1*R3P2)
LIK03140
N DEX=NOFST2 (C,N21,0, 10)
LIH03150
KSI (I NDEX) =X *Y1 *OMR2/ (R1P2* (01R3+P (3)*Y3))
LI 03160
KSI (INDEX+1)=KSI(INDEX)*73
LIM03170
GO TO 9999
LI03180
900 CONTINUE
LIM03190
C
LIM03200
C
------------------------------------------------ LI03210
C
LIMITING CASE NUMBER 9
LIH03220
C
(RIGHT EDGE AND CORNERS - XI GOES TO INFINITI)
LI1.03230
C
(CURVE: - + +)
LI103240
C
(NOCRALIZING: X1**N11 * Y2)
LIn03250
C
032-----------------Q3=O1l* (OlR1-O3P2*O+0P3*OaPIR 1)/(CO3P2*0 P1R 1* (ORE
1*OMR3LIE 03270
*
O11P2*OMP lE 1*01P3B3))
LIr03280
Z3=Oa i1/(Q3 *0 P2*OaiPla1)
LI03290
Y3= (Z3-0tiR3)/P (3)
LIM03300
-170-
Y1=-OR 1/P( 1)
LIŽ03310
X2=1.QO/Q3
LIM03320
C
…--…LIM03330
CALL ZEEO'(KSI,NSTATE)
LIH03340
INDEX=NOFST2(NSTOR (1),0,1,0,1)
LIM03350
L
KSI (I ND EX) =-2 :'f{OaB1*O*I?3-OMP t 1*0 1P22*0MP 3R3)/ (P (1)*R2P3)
LI M03360
IND3X=10FST 2 (N11,
,
0,0,1)
LIM03370
KSI (I NDEX) =X2* (0O1 1 '1OIR3/COP
1B 1B-0MP2*0MP3R3) /R2P3
LIM03330
KSI (IiDEX+4) =KSI (INDEX)*Y1
LI.03390
GO TO 9999
LIM03400
1000 CONTINUE
. LIM03410
C
.
LIM03420
C
0--------------------C
LIMITING CASE NUINBtR 10
LIM031440
C
(RIGHT EDGE AND CORNEES - X1 GOES TO INFINITY)
LI103450
C
(CUEVE: + - +)
LIM03460
C
(NORMALIZING: X1**N11Nt)
LIff03470
C-----------------------------------------------------------------------LI03480
LIM 034 90
Z3=0o P 2/(OMR 1 *OP2R2)
Q3= (CP3*Z3-OMP3R3)/ (Z3* (Z3-o1a3))
LI M03500
X2=1.QO/Q33
LIM03510
Y2=-R (2)/O3P2
LI M.03520
Y3= (Z 3-G. R3)/P (3)
LIM03530
C -3----------------------------------------------------;.__--- -LIm103540
CALL ZERO(KSI,NSTATE)
LIH03550
INDSX=3:OFST2 (NSTOR (1) ,C, 1,0, 1)
LIM03560
KSI (INDEX)=X2*Y2* (OMR 1*G1'R3-OMP 1R1*OMP2*CM3P3) / (P(1)*R2P3)
LI03570
INDEX=NOFST2 (Nl1,1,0,0,0)
LIMO03580
KSI (INDEX) =X2
LI 03590
KSI (INDEX+1)=X2*Y3
- LI103600
KS1 (INDEX+7)=X2*Y2* O R *Z3/P1P3
-LI 03610
IND3X=NOST2 (NSTOR (1) 11, O00)
LIM03620
DO 1001 N=1,N22
LIM03630
K SI (INDEX)=X2 **N*Y2* (O MR 1*X2/Z3 -OXP.1EP 1*C MP2)/R2P1
LI M036 40
KSI (INDEX 1) =KSI (INDEX) *Y3
LIM03650
1001
INDEX=INDEX+-8
LI M03660
INDEX=NOFST2 (N11,2,0, 0,0)
LIM03670
DO 1002 N=2,N22
LIO03680
KSI (INDEX)= X2**N
LIN03690
KS I (INDEX+1)=KSI (INDEX)*Y3
LIM03700
KSI (INDEX+6) =KSI (INDEX)*Y2*0:OR1/P (1)
LIM03710
KSI (INDEX+7)=KSI (INDEX+6)*Y3
L11103720
1002
INDEX=INDEX+8
LI 103730
INDZX=NOFST2(N11,N21,0,0,0)
LI03740
KSI(INDEX)=X2**N21
LI03750
KSI (iNDEX +3) =KSI (INDEX) *Y3 *O¶R2/P (2)
LIM 03760
KSI (INDLX+6) =KSI (INDEX)*Y 2*0OMR 1/P (1),
LIN03770
KSI (INDEX+7) =KSI (IN DEX+6) *Y3
LIn03780
INDZX=NOFST2 (NSTOR (1), N21, 1, 0, 0)
LIM03790
KSI (I NDEX)=X2**N21* (OMP1-O0R3*OMP1R1 *0
.2+.0MR2*R
1P3*Y3/R (3))/
LIM03800
*
(P(1)*(B (2)+R(3) -R2R3))
LIM038 10
KSI (INDEX +3)=X2**N21*Y3* (R (1) +R (3) -R laR3'R3P 1) *OMR2/(P (1) *R3P2)
LI M03820
INDEX=NOFST2 (N11, NSTOR (2),0, 1,0)
LIM03830
KSI(INDEX)=X2**N21*Y3*OMR2*01R3/R3P2
LIM03840
KSI (INDEX+4)=X2**N21*Y3* (R (1)+R (3)-RlR3) *OIR2/(P (1) *R3P2)
LI 03850
-171INDiiX= -OFST2 {NSTOR (l}, NSTOR (2), 1, ,0)
LI803860
KSI (INDX)-:
..
'2*N21*Y3'(R(1) +8 (3)-E1R3-{
r :;'1) *GOIR2* OMR3/(R*3P1*R3P2) LIi03870INDS,(=NOIST2 (N11 0,0, 0,1)
LI 03880
KSI (I I':.;X) =-X2 *Y 2*OLP2*OM P3R 3/H 2P3
LIM 03890
I ND:-X=NOFST2 (N12, NSTOR (2), 0, 1 ,0)
LIMl03900
KSI (INDEX) --- X2*IN21 Y3 -:;o1 P3*0 MP 2R2/R 3p2
LIt03910
GO 1'TO
9999
LIh03920
1100 CONTINUE
t11103930
C
LLIM03940
C ---------------------------------
L
03950
C
LIIIITING CASE NUMBER 11 '
LIM03960
C
(UPPER RIGtT COlNEl - Xl AND X2 GC TO INFINITY)
LI.03970
LIg03980
7
-+)
+
(CUP.E:"
C
C
(NORiHALIZING: Xl*+*N1 * X2**N21 * 1Y3)
LIM03990
C --------------------LI04000
Q2=O'RP.1(O'iR1 -OP20'-,
P3*O0P 1R 1)/(0iP3*CMPi a 1*(OM1*01R2LI1M') 4010
$
OaP3*O0MP1E 1*CaP2E2))
LI,04020
Z2=O01
Rl/(Q2*OMP3*OM P 1 ai
LIM04030
Y 1=-Our 1/P (1)
LIM04040
Y2= (Z2-OtIa2) /P (2)
LI U04050
C----------------------------------------------------------------------LIM04060
CALL ZERC(KSI,NSTATE)
LIH04070
I NDiX=NOFST2 (N12,NSTOR (2) ,0,1,0)
LIMO40d0
KSI (INDEX)= (OiR2*0R1 OMP 101-O1 P22 2*0MP3) /R3P2
L111a4090
KSI (I NDEX+4) =KSI (INDEX)*1I
LL104100
GO TO 9999
L110u41 10
1200 CONTINUE
LI11,04120
C
LIo04 130
C------------------------------------------------LI04140
C
LIMITING CASE NUMBER 12
LI104150
C
(UPPER i.IGHT COLNER - Xl AND X2 GO TO INFINITY)
LI04160
C
(CUrVE:
+
-)
C
(NORMALIZING: Xl**N11 * X2**N21)
C ----------------------------------------------------Z2=0z1P3/(0,'R 1*C P3R3)
Q2 =(0 a2*Z2 -o P22)/(Z2* (Z2-Oaa 2))
Y2= (Z2-021a2)/P (2)
Y3=-R(3)/O0P3
C------------------------------------CALL ZEO.(KSI,NSTATE)
INDEX=NOFST2(Nl,N21,0,0O,0)
KSI (I N3 ZX) = . QO0
KSI (1 NDEX +3)=Y3*OMR2/P (2)
KSI (INDEX+6)=Y2*CMR 1/P (1)
KSI(INDEX+7)=KSI (INDEX+6) *r3
INDiX=NOFST 2 (NSTOF (1), N21,1, 0, 0)
KS I(I NDEX)= (OaP1I-OR3*OMP1 E1*0Ml+E2+P. 2*1P3*l 3/ (3)
P ) / (P (1)*
·*
(h (2) +R (3)-R2R3))
KSI (INDkiX+3) =Y3* (R (1)+t (3)- R1B3-i 3P1) *CoR 2/(P (1)*R3P2)
INDEX= NOFST2 (N12,NSTO£ (2),0, 1,0)
KSI (INDEX)=-Y3*0 iP3*CMIP2B2/R3P2
INDEX=NOFST2 (N11 ,NSTOR (2) .0, 1,0)
KSI (INDEX)=Y3*0OR22*CAR 3/B3P2
KSI (I NDEX +4)=Y3* ( (1) +R (3)-RH l3)*O,1E2/(P (1) *R 3P2)
INDZX=NOFST2 (NSTOR (1), NSTCE(2), 1,1,0)
LI04-170
LIt04180
IM04190
LIM04200
LIMO04210
LIM04220
LI 104230
LI04240
LIM04250
LIM04260
LIM4C270
LI ,04280
LIMO.4290
LIM04300
LIM04310
LIM04320
LIo04330
LI104340
LIM04350
LIM04360
LIH04370
L1I104380
LI%04390
LI104400
-172KSI (I iT'?bX) =Y3* (I. (1) +R (3) -Rl1R3-R3P 1) '0,.920,*0 R3/(R3P l*R3P2)
I;i)':X-=NoFSr2 (NJ'£OR (1) ,N22,1,0,0)
KSI (INDEX) =Y2*OMk 1/( (O
IC13+P (3) *Y3) *R2P1)
K SI (INDEX+1)=KSI (INDEX) * 3
C
9999 RETURN
C
C t***t END OF LIKKSI
C
END
LIM04410
LIlO 4420
LIM04430
LIM04440
LIMO4450
LIM04460
LIM04470
LU1104480
LIMO4490
LI0O4500
-173S Itj3CU'i JN
LIIIVAL i(:CASE ,il,
LI1100010
LIu0v)20
ST2, ST3, S%4)
C
----
C----------------------------------------------------------------LIMITING KSI-VECTOR EiirdIES
COŽ.ilT.S THi
THIS SUBiRuUTIN
C
TO
r'iiE FOUii F LLVUtNT STATES IJTILIZIED BY
CORhiSPONDIIi;
C
tliNGULAh VECTOR.
C
'SCAN*' TO. OETLIEMINE TilE COr(JEiC
(LIMVAL iAML) (LIdKSI NAnm)
C
KSI '4)
ST1
C
KSI(INDiiX*1) , LiLO.iX=ONOFS2 (0,1,03,1,0
ST2
C
X'1iiX),
IND'X=N;CFST2'NSTO fi1),fSTORfl'2) 121,1)
KSI
S'i3
C
KS1(iNDLX+7), INDEX=NFS'T2 (N11,N21,0,0,0)
ST4
C
C
.'O)
KSII.NDLX+4~), 1DEX=;GFST2 (N11,,21,0,1, 1)
ST4
C
C-----------------------------LIM00
C
2
P3,
3,
,0t13.Pl,02,01"?
,2,0m3
:1
3,0MI
CO;:ICN/PACi'O0R/fi1, 2.2,t3, PlP2
3E3, E1 .;2, R1E3, 1P1, 31P2, 123,; 2R 3,-. 2P 1
*
C 1 1 ,CM22,
2P3, N1 ,N2 ,N11 ,N21,
i12P2, .2i3,.3 £1 , r.3P2,13P3 ,P1 P ,I21 ?3 ,i'
*
*
LIM0006'J
LIiOU070
LIiOJJ080
9
LId
00
LI0l1 )0
LIiOo11
LIMtJv120
LIiiJv130
140
LIIM00150
Lt410 160
LI(itC 17')
LIM0I)180
LIM0j 190
N12,N22,;NDIF
2)
CO IMON /PA n A1S/ '3) ,P'3), .3NAX
C
IN rEGEl
INTEGi.L
L IoQij3G
LIHM00 40
LIi13v050
N11,N12,N21,;N22,1PDIFF
N1, IN2, NCAS1:, INMA
:
LI 100 '200
LI11, 210
LIIM;0020
LIa00230
LIiMJO240
C
,BR,P
,P ,'33
,3,1
2
rL'AL*16 ST1, S'EP2 S3SP
1.I
,X2,Y1,Y2,Y3,Z 1,Z 2,Z3, Q1,Q'2,Q 3,
*
2,C23 F.3
C.P3,C MP l 1 ,
O2,
_*Mhl, C 2, 0 M3, O3,,P1,OAP2FeL
R lhi2 ,t R 3 E 1P 1,R1P2, 1P3, E2.2i 3,I2P 1, R2'2, R2P 3,R3P 1
*
R3P2,i3P3, P1P2,i 1 P3,P223
*
C
GO TO 10C,20,30,40,50,60,70,80,93,
100,110,120) ,UCASE
C
10 CONTINUE
3 * (OM.i3*O[i21
Q2=C Mh 3 (0.133-cM ?1 o0P2 *0 i.P 3r.3) / (OYP .*(P3R
*
QOYP1*4OMP3R3*OiP2R2))
-LI
Z 2=0 1M3/( Q2 *Ct1*OP3R3)
Y2= :Z2-OMH.2)/P2
……LIM
a
--------------------C-----------------------ST1=Oii, 1* (h 1+R3-E 1.* 3-i 1*P3) *Y2/ 't1*R1*P3)
ST2= (i· 1E3-i 1*Li 3) *Y2/ 'R l*P3)
ST3=0.OQ +0
=
. OQ+O
ST40
GO TO 999
c
20 CONTINUE
Y =-R 1/C?21
Z 2=CGP1/(O01P1a1*OMRi3)
Y2= 'Z2-CMR2)/P2
-0
C… ------------------------ST1=Otii 1*'i,1+h3- 1 *i3-F.1*P3) *Y 1*Y2/'.R I*R 1*?3)
.LI.SOi510
ST2= 'it1 +R3-l 1 *fi3) *Y 1 *2/[R1 *P3)
ST3=O. u-.,+0
+4
ST=).GO TO 999
C
LIH00250
L15030260
LIZ00270
LIMa30284
LI 0tUo290
LIMu03;vO
LIM00310
LI00320
LIM00330
LI.300340
LImc0t350
M00360
LIS00370
3O
LI O u380
LIa00390
Lli1O0400
LILI04 10
LIMON 420
LIMOiG430
LIIMOu440
LI 100 450
LIM0046 0
LI O)0,470
LIM00480
LI490
LIaLO500
LIM00520
LIMO0530
LIdOOS40
LIO00550
-17430) CI)NTINUL
2
),2*IQ)R3) / (O iL'1'b. P2R2* .2 (
3 C M . (Ci:!.2- C 1P1*O:i P3J'G I 2 hR
i.'1i:ot u2H212'O;sIR3)3 )
+
R2)
Z 3-C l1 2/( 3 *Ci I?1*GOaP2
t
Y3= {Z3-O3R3)/P3
Y2=-CMEL2/P2
X 2= 1. 0JQ+0/Q(3
-------C --------------------------------------------*
X2*t2/ (ii1*1*P3)
t 3-i;1P3)
ST 1=Ciii t1 1
R,1*P3) .
9'T2- ' 1 +R3-R1*tk 3) *X2 *Y2/
.
ST3=. 0+0
ST4= 3O.0+3
GO TO 999
C
40 COiNTINU
C---------------------STr 1=0. 0% +O
.O
S12=0. :)
ST3 =3. 2.+0
ST4= 0.0+ 3
GO TO 999
..
C
50 CONTINU
tOMa. 10,o.*3M3-OdP2 *OM1i El *10MP3Ii3)/'iKP2*GiaP3t 3*
Q1 =C.ii3 * tUo
1) )
.
*
Gd P2* oP31,3*o0i?la
Z 1=O0ah3/ ,-1*CaFP2 o0 P3 R3)
.
X 1=1
.1
Yl= :TZ1--oa1)/1
Y3=-OMh3/23
C- ---------------------------------------------------------'
1
3-EI. 1*,P33)* i l * Y1l/' E1*R l*P3)
,1+E
ST1=OM?1*r
X1*Y1/(1*P3)
ST2= (l1 -I3-R1*3)
.
ST3=0.0Q+0
ST4=0.02+0
GO TO 999
C
.
60 CCNTINUE
2OM r 3)
Z 1=Oi?2/ 'Oa 2 2*
.
1)
I(Z
/
1 :Z 1-0D5
l
AP1 1)
01 =:C;1£1 z1-O
X 1=Q1
Y1= Z1-C,HR 1)/P1Y2=-R2/CiR22
C--------------------------------------Y1 Y2/ ( 1*R1*P3)
ST1=Cil*'tIR1+Z3-Rl *3-1 1.*3)*1 Yl
ST2 = ' 1. 1+R 3-k 1*t: 3) *X 1 *Y1 *Y 2/: 1 *p3)
ST3=0.03+0 .
ST4=0.0 +0
GO TO 999
C
70 CONTINUS
(O !3*oiIP22*. (0oalt*oIR2dP3 *oiP1 *0.P2P2)/:
l1=o0iaŽ *toaa2-C
·
*
oiP3*0 32a2*0
oP 1R 1))
Z 1=OAR2/ fQ1*CHP3*OM P2R2)
1= Q1
YI= 'ZI-OIH 1)/P1
LII00OU50b
Llih0 570
LI;tuJ580S
LIM0 3O
590
LI 10 00)
LI;300610
LI M00 620
LI)0630
Lito4
IiH3-Oi1
Lii:J,650j
LI i1O0uG60
LI iiJj 670
Li(3. 68-J
LI'(0 0b90
LIii0u700
l.
IMOf100'1
LIM0U) 720
LIa (-.73
LI 3dO740
LIMi0?750
LIMv 760
LIM00770
LIM00730
LIMJl3790
Lii0C 30
rIMOO810
LIM00820
LIMv,,,839
O40
LI3
LIMO0 850
LIM09860
LIMI0ocU7
L.lXJ880
LI.0O 890
LIMOi900LIM0091 0
LIMO0920
LI?.f9,930
LI 00940
LIO00950
LIM0 96 0
LI 0O970
LIav;98&
LIM06990
LIPi3G 1 0
LIM01010
LIMO 1020
LIHv1' 030
LIiO 1o40
LIM0150
LIaO1 060
LUi0 1070
LIM01 G80
LII11090
LIrt1 130
-175Y2=-C Lk Z/P2
C ------------------------STl=0.0~+0
-.
ST2=0.++J
ST3=X 10* (.N 1- 1) * Z* (±I1 +i 3-*
/ I'/YiE *1* 13*P2*R3)
ST4=X 1** ( d1-1) *Y2*4Z 1/-1
GO TO 999
C
....
80 CON TIN OZ
Z 1=0MP3/ ?O1i,2 C P3R3)}
1I= (uaiPI*zX 1-3M P 1 1) / (Z1* (l-Ol}
X 1=Q1Y 1= (l-Cmd 1) /El
.
Y3=-i3/0a
13
R3*P1 ) *iiOd.2*Oa3
.
.LI01
}
.
-
-
C-----------
LII01260
ST1= 0. 0+0
ST2 =0. 0iQ+)
.
S-T 3= X1 - (1 - 1) *y 3*Z 1* (a 1 +2t3 - *R
1 3-h3. P1I*0MR2
)
*0 PR3/
*Oi 1 *R3*P1 *Ai3*P2)
S t4= 0.0°+0
G; TO 999
iId 1270
LI 301 80
LI;Hi 1290
1.IM0)13 00
LI~iU 1310
LIi1l 320
LI 0 1330
LI101340
1
5--------------0
LIM01360
LIM1)370
LIM 01380
LI 0 1390
LIM14Q00
LIM101410
LIX.01420
LIdOl 430
Lza 1440
C
90 CON TIiN US
C----~-----------------,,,
,-,,-,,-,
--- ,--,,C---…LIM01350
ST1= 0.0+
ST2=0. Q+0
ST3=0.0OQ+
ST4=0.+O+
GO TO 999
C
100 CONTIN UE
Z3=Or;P2/ (OMh1 *C P2 2)
SQ3= ( C,1P3*23-c, 3R33) / Z3* (3-oR3) )
X2= 1 . i,+3/Q3
Y2=-R 2/C3P2
Y3= 'Z3-Oa1h3)/P3
C
LIift11 0
LIAO 1 120
LitJ130
LI01 140
LirA0 1 150
l10
L;1a 1t170
LIi51l 18J
LIMI 190
LIi0 1200
LI1012210
LIM01220
LI: )0123 0
LiV)1240
LIM a1 250
----
,---,-,,
-
.LIM01450
…------------.---------.--.
ST1=0. 0,+0
ST2 =0.) .+0
ST3=X 2** ( N2-1) * Y3* (aR +a3-h 1*R3-1a3*eP1) *0if2*OHE3/ (3 *P1 *-R3*22}
ST4=X2*
N2- 1) *Y 2*Y 3*0 h1/P1
GO TO 999
C
110 CONTINUE
.
IC……------------------------------------· ------~~~-------ST1=O f.
0+90
ST2=0.0Q+0
ST3=0. OQ+O
ST4=O.C2+0
GO TC 999
C
120 CONTINUE
Z2=OaMP3/:
01a1*CP3R3)
2= (Ch P2 *Z 2-C AP2R2) /(L2*
(Z2 -O a2 ) )
-
LIM 01460
LIiJU1470
L1101480
LIh0149
LIO1 500
LIn01510
LIA01 52 0
LIa01530
.LIaO1540
L101550
LIL
.01 560
L Ia01. 570
L1.0 15b0
LIM01590
LIX01600
LIO 1610
L1a01620
LI0 Olb30
LI:U 1640
LI0 1650
-176Y 2= (Z2-Oi 2J /P 2
Y3=-£3/CM23
C........--
-
ST1=O. C~,+.~
SI2=UO.+0
ST3= Y3* [ 1+i- 3-ft 1* 3-ar3*P1) *Ot'h2'071RI3/ '~,*u1*Efi3*PL2)
STQ=Y2 *Y3 #C;RI/P1
~~~C~~~~~~~~~~~
999- RETURN
C
C **** EID- Of LIIVAL
ENC
~~~~~LIAU
-
LIAO 1660
LIMO1670
LI
lbdO
LIM0169'~
LIHO 1700
Lla')1710
LI nO 1720
1739
LIM01'~O
L;1I, 1750
LIifti
76 )
LItAO17d7
-177-
STOO.I1 a
IALPHA)
INTEGIk FUNCTIGN OGFST .N,
STGOO020
C
3
C----------------TO---------------------------------------------------T
STcQOOO40
OF T}I FOR3
C
THIS SUi.ROUi.'INt CONVLRTS A SYSUAI STATeG
STOL3)t,
),
LALA3)
'iN (
), a '2) , ILi
C
ST00u060
iOWOF. COLJaN ihDLX IN THE'TRANSITIO1
C
INiir
IT.; CORtiR:SPONDi)NG
STOO070
MATRIX Old PROiAUILLIY VECIORi.
C
STO(uS8d3
C
STO00090'
C
THE AF.GiJ:I.NTS ARL GIVEN IN AF.fAY FCRa.
STO, 1OU
(EXPLICIT LISTIa'G OF A.lGiJl;.N'S RAiTHER THiAi Ait:AY: SE1Z NOrST2)
C
C
-----------------------------------ST00110
STOOu 120
C
STOi130
CO aCON /PAihAIS/ h'i3), P'3), N.IAX{2. .
STC00 140
C
STOwu150
TNTE(;ik R 2), iALPHA 3)
ST000160
.
INTEGER dAX
SS-re00s7-3
roo;o17
Cc
..
STO0C 18u
R iAL*16 R,;P
STC0' 190
C
ST00C200
IALPHA(1) +
= 1 + IAlLPHA 3) + 2 * IALPdA.2) + 4
NOFS
STOO00210
8b*
(iN(2) + {(1) * (IiAX'(2) + 1))
*
STOOu220
RETURN
T000 230
C
STOOOi240
C ***** END 0? NOFST
ST000250
C
ST000260
END
-178-
INTEGER FUNCTION NOFST2(tl,N2,1A1,IA2,IA3)
…C
~~~~~~~~NOF00020
C--------------------------------------------------------------------THIS SUBROUTINE CONVERTS A SYSTEM STATE OF THE FORK
C
IALPHA (1), IALPHA (2), IALPHA (3))
(N (1 ) , N (2),
C
C
C
INTO ITS CORHESPONDING ROW OR COLUMN INDEX IN THE TRANSITION
MATRIX OR PROBABILITY VECTOR.
C
THIS IS
~~~~C~~~~~~~~~~~~~
A VERSION
OF 'NOFST'
WHEREIN
THE ARGUMENTS
ARL LISTED EXPLICITELY RATHER THAN GIVEN AS ARRAYS.
C
---------------------------------C-------------------------------C
COMMON /PARAMS/ R(3), P(3) , NMAX(2)
~~~~~~~~~~~~~c
REAL*16 i,
P
NOFST2 = 1 + IA3 + 2 * IA2 + 4 * IA1 +
+ 1))
8 * (N2 + N1 * (NMAX(2)
*
C
C *****
END OF NOFST2
C
RETURN
END
NOO30
NOFO0040
NOF00050
NOF00060
NOF00070
NOFOC080
NOF00090
NOP00100
NOF00110
NOF00120
NOF00130
~~~~NOF00140
INTEGER N1,N2,IA1,IA2,IA3
INTEGER NMAX
C
NOF00010
NOFO0150
NOF00160
NOF00170
NOFOO180
NOF00 190
NOF00200
NOF00210
NOF00220
NOFO00230
NOF00240
NOF00250
NOF00260
-179-
SUBROUTINE
C
NTRAN (A,B, NI,2,III)
.,I
01080
PMId1,9
C
C
C
P1
01-100
------------------P2MI01110
THIS SUd3ROUTINE COMPUT6S TIlE FINAL STORAGE LEVEL
GIVEN T.El INITIAL STORA:E LEVELS AIiD TIHE FINAL :3ACI{INE OPERArIDNALP-11I1129
PIi01 130
CONDITIONS.
PI01 1143
_
TRHE SrARV3D AND BLOCKED CONDITONIS
ARE TAKEN
INTO ACCO]JNr
PaI01150
PM101160
BY FUNCTIDNS 'IU' AND 'ID'.
C
C…-----------------------------------------------------------------------PIO1170
PM2I01183
MODIFIL) TC ACCOUJT POR STORAGE BACKUP
PMI0r1 190
C
PMI01200
o.0O,0N /NS/ I1N(2)
PMI11210
C
PMI31223
INTEGER A, B, N1,.N2, III,IN, IU,ID
2PI01230
+
N2 = N1
PMI01240
(B-1)*ID(III)
(,-1}*IU {II)
i1
PNI0125,
RETURN
P1IO 1 260
C
PIO 01270
C ***c* END OF NTRAN
P I01280
C
PMI0 1290
END
-180-
ATPO0540
SUBROUTINE NTR.lNS(NP~f.V, NNEW, ALPHA1, ALPHA2, NIACH)
ATF00550
C
----------------- ATE00560
--------------C.---- ----------5-----------ATE00570
TIHIS SUBROUTIINE COMPUTES THE FINAL STCRAGE LEVEL, 'NNEW'..
C
ATR0o580
GIVEN TilE INITIAL LEVZL, 'NREV', AND THE STATES OF ADJACENT
C
ATR00590
MACHINES.
C
WHETHER OR 'NOT THESE MACHIINES ARE STARVED OR BLOCKED IS TAKEN INTOATE00600
C
ATR00610
ACCOUNT BY FUNCTION 'ITRANS'.
C
- ---------------------------------------- ATE0620
C------------------------ATR 00630
·
C
ATR00640
INTEGER NPREV, NNEW, ALPHA1, ALPHA2, NMACH, ITRANS
ATRO0650
C
ATR00660
NNEW = NPREV + ALPHAl * ITRANS(NM¶ACH) - ALPHA2 * ITRANS(NhACHt1)
1TR00670
iETUDN
ATR00680
C
ATR00690
C ***** END OF NTRANS
ATRO0700
C
ATR00710
END
-181-
SUBEiOJTLNE PdINAP P, A?, NAP)
C
.
C---P..............PNOOO3))3
C
THiIS SU31ROUTILNE COI:PUTS TIHL iihOh
P
T * e
C
C
([IlRa . P=P1.C3AbILITY V' CTOR, T='IrANSITIUo
C
CAUS'cD J1Y Pf LCISIC'N / XOU lDOFF PROBLEMlS.
C
"
.
C
ANALYTICALLY, P = T * P
C
.i3jl100
;
C
THIS SUR3OiUII1N. CALLS
IAl.S A AND 'JNT. AN .
C
'INTRAN' CA.LLS 'U I aIND
;
AID.
C
C~dMCOi /2AL.,AM3S/
t?.IO)10
-100020
1P?I00 40
PMIi, u5)
idIOO 110
PHIAL,123
--------------------------------------.
'.f3), P?,3), NP (2.)
COaON /NS/ J.41, J32
.1IJU
B
A1,U
A2,B2,
1
iNTEGER AA, BE1,
INt(£GI
:,.NP
kJill, :1N2,
NNP1, 1i2l, K,13, I,
· .
JN1, .JN2,IN1. IN2, J1, J2, J3
C
C
PlIOuObO
PA IO C70
PflOJJ8O
I. ju90
NATHIX)
I1,
P
~-I
Pli150
1Mli,'16'3
170
PlO0'180
d31 0019O
12,
PdIOO200
P10l0210
.
P(NAP), AP(NAP),, , P?, £E. 2, PP1,
AX, SUAD1F, DIFMAX, DIF, QAS
,
I
NN11= 1= :1)
NN21=NP (2)+1
K=N N11*lN 21* 8
tlAL*16
PP2,
P.IO0U220
I
PP3,
*
C
.R 2=1.Q-6
90
8008
8007
8006
DO .90 I=1, MAP
AP(I) =
CONTIUNUE
DO 8001 I1 = 1,2 ' ·
DO 8002 12 = 1,2 DO 8003 JN1 = 1,_NI11.
DO 8094 JN2 = 1,NN21
CALL lITRAN (I1,I2,JN1,II,1)
DO 80C,5 13 = 1,2
(I2,13,JN2,IN2,2)
CALL 1lTdAA
.
DO 8006 J1 = 1,2
1 ,JN1,J2)CALL AIRAN (1,J1,I1,PP
ARR2) GO TC 8506
IF
2P1 .LT.
DO 8007 J2 = 1,2
2 ,J2, I2,PP2,JN1,JN2)
CALL ATRAil
IF (PP2 .LT. ERlF2) .GO TC 8007
DO 8008 J3 = 1,2
2)
CALL ATRAN (3,J3,I3,PP3,JN1,J3N
AX
= 2P* PP2*PP3
.RR2) GO TO 68)8
IF 'AX .LT.
A2 = I3+2*I2+4*I1+3*1N2+8*NN21* (Ib1-1)-14
* (JM1-1)
lI0O
-14
A 1 = J3+2*J2+4*J1+6*JN2+8*NN21*
IF AI.JlT.K.CR.A2.GT.K) GO TO 8B08
AP(a2) = AP(A2) + AX * P'1l)
CONTINUE
CONTINUE
CONTINUS
30
14U
,PM100240
PMI*;j250
PHI00260
P1110 270
.
-P10I0280
-
PMI00290
2,1+0
PMI14C300
Pl2I00310
320
PI
P2I00330
PMI10#340
PSI0Iv350
PI100360
PHlJ0370
PI 00380
PI390
PnIOC 400
PuiO41 0
PHI00420
P3l1i430.
PH100440
PSO16 450
PH130460
Pa10O470
PMIcL-480
PMIU0490
0
PI00510
P100u520
2PI0O530
?PIO0 540
PO110550
-182-
8005
800t4
8003
8002
8001
CONTINU i
CONTINUZ
CONTINUE
CCNTIIU
CONT lNU
SUMDIF = 0. 0+0
DIFHAX = )..)Q+Q
DO 1iCO =I,IAP
DIF = P 'I) SU&DLIF
AP 'I)
= SUAtDI1
+ QABS1DPIF)
IF (DIaiiXt
. LT. ,'2ALjS (DL)) DlFMAX = QAB:'iDIF)
100 COiNTiU
'
W'IT( .(6,90.)) SUMDL1 , CIFiiAX
900 FOLE.niT '/, 'i'i
SUti OF 'iit
ABSOLUTr; VALrJZS O(F P
*
/,'
T'H'
i 1.AX iilU1 ELr.A EN T 0,F Ti lIE;F
HETURN
C
C ***** END O? P?1NAP
C
END
PMIO00 560
Pii i .570
?2I00580
2?MI0 590
PMI0o600
iI00610
d
£'I)'.
620
PMIOub30
a i),640
PMIu.io50
AP WAS ',213.6,
WIAS ',Q13.6)
PIuU0660
PI uu670
PiM100680
iPlHIO690
ilI031700
2 IAtl7 1 J
P1d;)0720
P KeIOC,73i)
P MIO 740
-183-
SUBROUTINE PRINT(STATE,
NSTATE)
C
C
PRINTS PROBABILITY DISTXIBUTICN WITH THREE-STAGE LINE
C
FORIAT.
C----------------------------------------C
COMMON /P.%RAMS/
R (3), P(3) * NSTOk (2)
C
INTEGER NSTATENSTOR,N1,N2, NBEGN,I,NN1,NN2,K
C
REkL*16 A, P, STATE(NSTATB)
C
WRITE (90,1) NSTCR(1), NSTOR (2), ( (I),I=1,3), (P(I),1=1,3)
1 FORMAT(1tl1,2X,'PhOBABILITY DISTRiBUTION FOR N1 AND N2 = ',2I5,/,
*
'
PROBABILITIES OF REPAIR = ',3(FB.6,2X),/,
*
'
PROBABILITIES OF FAILURE = ',3(F8.6,2X),//)
N 1=NSTOR(1)+1
N2=NSTO'R(2) +1
NBEGN=O
DO 6 I=1,N1
NN2=0
NN1=I-1
WRITE(90,2) NN1
2 FOEMAT(lI[0,/,10X,'N1 = ',I5,/,3X,'C00O,1CX,'001',10X,'010',10X,
1
'011',10X,'1000,10X,'101',10X,'110,10X,'111',21X,'N2',/)
DO 6 J=1,N2
WRITE(90
,5) (STATE(NBEGN+K),K=1,8),NN2
5 FORMAT(1H ,dE13.5,11X,I4)
NN2=NN2+1
6 NBEGN=NBEGN+8
RETUE N
C
C ***** END OF PRINT
C
END
PRI00010
PRI00020
P-------------------------------------RI0
PRIOC040
PRI00050
PRIO060
PRI00070
PRI00080
PRI00090
PR100100
PRI00110
PRI00120
PRI00130
PFI00140
PRI00150
PRIOC0160
PRI00170
PRI00180
PR100 190
PRI00200
PRI00210
PRI00220
PRIO0230
PRI00240
P:I00250
PRI00260
PRI00270
PRI00280
PRI00290
PRI00300
PRIO0310
PRI00320
PRIO0330
PRI00340
PRI00350
PRI00360
-184SiJBROUUYIN.: USOLV A, B, C, SOL1, SOL2, -IFLAG)
C --------------------C
.
(OTS
'rs OL1' AND $SOLz
COitP'JTS TiHE T'O
C
TItiS SU;3ROUTINEi
C
OF Tiif e QUADRAIIC
QIUATION
+ C
= 4
* B * X
A * X**2
C
IFLAG=-- IF ROGTS Air C0(i2PLEX (REAL ?iA'rT=OCL1, i.AG.XitTP.=SO,2)
C
;S0001*
20
-SO'J&030
QSO0oJ40
2SOG00050
2SUO'`'J360
i0t,;070
sG
...........................
iNTEGEi
SLou )~0}
IFLAG
C
QSGou103
SOOI
0110
QSO -A- 120
lSO0u130
QSOi0t, 1 40
dSOG.i 15)
uSOGO'i 160
:$, S17.)
ISC0j 180
VS.iuv 190
.
REAL* 16 A, !3, C, O L 1, SC L2
RhEAL*1S
ROOT, :SUT
C
iF LAG = 3
RGOOT = J * B - 4
.. L..
IF
'1ROT .LT. i0.J.+0)
*' A * C
GO TO. 100
C
RO2:T = jSQEHT(GOT)
IF (B .GT.- J).,+0)
LOCr = - LOOT
SOL1 = (-S + iOCOT) / (2.,O+O * A)
SOL2 = C / fSCL1 * A)
hEI.I U
QSO0U200
uSO00 210
2SOOC22)z
SOiu 230
SiU -i:2 q40
SOU 250
i SOOt.26)0
270
-
C
100 CONTINJIE
IFLAG = -1
SOL1 = -b / (2.3Q+O * A)
.SCIUu
SCL2 = -SQET '-ROOT) /
2.0Q+0 * A)
hETURN
C
C ***** END Of
C
EnD.
£
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2S.OLV
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S000310
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SUi.OUTINE lSCAtiW (W, B,NeLDD,U,NOD1D,INDXVL,INDtXW)
C
C
C
C
C
C
C
C
C
SCAOO0 13
SCAOt)02 0
.
.
..-..
.......
----------------CA030
THIIS SUt}hLOUTIZNL ATriSAPTS TlE 'C-SCANNiNGI' DETESiLNATIO;
OF TIiE CORiiirECT SIN iLAEF V£CIG;UR TO CiTAIN THE
-POEBAbILITY VzCTOiH F.OL1:
P
=
SUdO(
C
*
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3
}
J
J
J
'.{IIERE '.L IS Tli. SI(;GULArt
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SCAvC,8.
SCAOUi090
SCAOiJ 10J
C
f dE
SI:iGULAh V.;CTO[i a;hlCii YILLDS Tti.
C
C
C
CF TilL
iodABi3ILITIS Of FOURA FEOQUNT STATES iS SELECTED
-AiMGNG T6l03r. COWi~kS'UiND1I;G TO SINGULAR VALUSS BZLOW :ACiIINi
PI.PCISICI 1.ANGf.
c
-
COt;14ON /i.Ai'Ar
SUN
SCAO0110
SCAUu120
SCA Ou13
SCAOUv140
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S/h '3),P '3), N:AX'2)
SCAOJ 170
SCAO0 180
IN-EGiE•
I:JDEXW (UFODD) ,iCDD,;)iiOD)1 ,AGAIN, ID.IY, N2XP, IDLST, 1 DXVL, SCA00190
·
N1,N2
.
SCA00200
SCA0U 210
E LA L* 15 X 1 , 2,Y 1 ,Y2,Y 3, P 1,P2,P 3, R1, 2,3,
SCA00220
·
'Gi30 1 1 , P 01 01
1,iD, iUr,
O , TO PX,DGTE ,TSUi;
GI
j,
SITSMDUIYr
SCAJtu230
·
*
d 'vNFC3DD),3 ' NFODD, NFCL)), U N fODiD, 5) ,CUTO fF,R,P
SCAuv,240
i{LAL*16 CAR
:1,CiZi.2, Oi'E3
.CA30250
S CAuLt26u'
SCAOU270
PI=P1 )
SCAOJ 280
P2= P(2)
SCA00290
P3=P 13)
SCA00300
1=R(1)
s cAG313
R 2=: [2)
SCAOi 320
R3=R {3)
SCAO0 330
N 1= N1'I
AX (1)
SCAOi
N2=N3Ax '2)
4 ,340
SCA0 0350
OI 1:
1 .X OQ+O-1a
0 a 2=1. ,U)+- R2
S CA(ro360
oMR 3=1. 3 +0-3
83CA0o370
c
C
C
10
DO 10 I =1,NODD
INDEXWi 'I)=I
c
NOD 1= NO DI)- 1
15 :AGAIN=.)
DO 23 I=I,NOLD1
:I+1))GC TO 20
IF:'I).G.W
U
,I)
*
DyIy
w I))=4 :I+1)
=IJuily
)
w I+1
IDUoiZ Y= L1i[DNi.Ai fI)
:I-1)
INDOXi
:I)=INl;EXi
IN D.X ,X4( + 1) = I DU Z
IAGA1J1=
20
CCNTI.Ug
IF 'IAGAIN.LQ. 1) GO TO 15
C
LARG--.T-YA.GNITUD'::
SCA00390
SCAQ(,400
SCA 00410
SCAOU 420
SCAJ0430
SCAOU440
SCAuis45O
SCA0 460
s CA;L'470
SCA04 80
SCAJV 490
SCAJ30500
sCAO50
iO
SCA 00520
SCAO 530
SCAu 50
$cA00550
-186-
NEXP=QLOG10 (W(1))
CUTOFF=10.0** (NLXP-32)
BESTSM=O. OQ+0
IBEST=O
C
25
C
26
27
30
DO 30 I=1,NODD
IF(W (I).LT. CUTOFF) GO TO 25
IF((I.EQ.NOLD) .AND. (IBEST.EQ.O0))GO 10 40
GO TO 30
TSUM=O.OQ+O
DO 26 J=13,NODC
X=U (J,1)
X2=U J.,2)
Yi=U (J,3)
2-U (J,4)
Y3=U (J, 5)
P00011=X1*X2*Y1* Y2*O R 1* (R1+R3-Ri*R3-R1*P3)/ (P3*RI**2)
PO 10 11=X1*X 2*Yl*Y2* (RP1 +3+R1*13) / (Rl*P3)
PTHIRD=PR(N1 N2 1 1 0)
PFOUr,=PR(Ni-1 N2-1
1 1 1)
TOPEX= (X1** ({N1-1 ) * (X2** (N2-1)) *Y3* (CIR1 +P1 *Y1)
BOTEX=CR1 *Pl*F2*R3**2
PTHIED-=TOPEX*OHR2*OR3* (RP+R3R*R3*R3-R3*P
)/BTEX
PFOUR= (X 1+*(N 1- 1) ) (X2**(N2- 1) ) *Y2*Y3* (OR1+P1*Y 1) /P
TSUM=TSU+ (B (J,INDEXW (I) )* (P00011+P010 11+PTHIRD+PFCUR))
DO 27 J=1,12
CALL LI1VAL(J,P00011,PO101,PTHIRD,PFOUR)
TSUi=TSUI+ (13(J,INDEXW(I))*(P00i11+P01011+PTHIRD+PFOUR))
TSUM=QABS (ISUM)
IF(TSUM.LE.BESTSM)GO TO 30
BES1Sl=7SUM
IBEST=I
CONTINUE
C
INDXVL=INDEXW (IBEST)
GO TO 50
C
40
45
900
50
WRITE (6,900)
STOP
FORtiAT(' SINGULAR VALUES NOT SHALL ENOUGH FOR SCANW
CONTINUE
RETURN
C
C ***** END OF SCANW
C
END
')
SCA00560
SCAOO 570
SCA00580
SCA00590
SCA00600
SCA00610
SCAC3620
SCA00630
SCAO0640
SCA00650
SCAOO 660
SCA00670
SCA00680
SCA00690
SCAO0700
SCA007 10
SCA00720
SCA00730
SCA00740
SCA00750
SCA00760
SCA00770
SCA00780
SCA00790
SCAO0800
SCA00810
SCA00820
SCA00830
SCAO084O
SCA00850
SCA00860
SCA00870
SCAOO 88O
SCA00890
SCA00900
SCAO0910
SCA00920
SCA00930
SCA00940
SCAO0950
SCA00960
SCA00970
SCAO0900
SCA00990
SCAO1000
-187SURCUl3iT.l.
STOFNil
(NS£AT, N, IA'LPIIA)
sTOviu300
STO00310
C
C----------------------------------------------------------
C
C
-----------
THlIS SL.)sOUTI:zE CON Vr.iTS TliE ,OW Ori CGLUt.N IN)LOX OF A STATi
!N'TO Ti?.: ?ORa :
C
( (1),
J(2),
IALiIA (1),
IALPHA(2),
IALPHA (3))
C
C
g NSTAT
IS T.iE i.On/COLUMIN INDLX (F THti STATE.
C--------------------------------------.
C
colai./?AniAAS/ i ,(3),
2 3),
LIIAX 2)
c
lNTEG;Eh .ISTA'T, I'TA.IE, N(2),
IALPiiA(3)
Uzi2
INTEG1.k. JNAX
INTLG. it I IT,
Ur,
C
R'.AL*16 ii, P
C
NSTATE =- STAT - 1
OD(:{STAT
.
2)
IALPHlA(3) =
NSTATE = IISTATE / 2
IALLPiA(2) = 1MOI(N.STATt,2)
NSTAT. = NSTATL / 2
IALPHA(1) = .40i(NiSrATE,2)
NSTATE = NSTATIB / 2
N.i2 =. AAi 2) + 1
a(2) = 20O(NSl!ATE, N 2)
N 1) = NSATE / N2
c
RETURN
C
C****
C
END OF STOPN
END
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o
2
STO00330
STOO;i34u
STO00350
srobA36J
STOOU370
380
ST000390
ST(0r. 4u;J
ST'OOu410
STC1iU1420
srooo
620
S'TOOj43 3
ST0v0440
STOj, ~450
STOu0C460
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ST000I48
STCGO 490
ST03u500
STOuS510
STO 0520
TOOC530
SIOO054 0
ST000550
ST000 560
sroT000570
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ST0OO0590
ST000600
STON
T)0610
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ST000630
SUiiOUTI'..
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OUT Tifi iU'
AND 'C'
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-----
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REAL*16 A, J, C;
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B = X *
iX
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2+o
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C
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FO..1S THiE QUADRaATIC E2iIATIOi k'Oi 'Y3' 1N
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RE-AL*16 X, Y31, 332
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C ***** END OF Y3SCL
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Y3S00220
Y3S0U230
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Y3S00250
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Y3SUoi270
Y3S00280
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REFERENCES
Anderson, D.R. [19681, "Transient and Steady-State Minimum Cost In-Process
Inventory Capacities for Production Lines", unpublished Ph.D. Thesis,
Department of Industrial Engineering, Purdue University.
Anderson, D.R. and Moodie, D.C. [1969], "Optimal Buffer Storage Capacity
in Production Line Systems", Int. J. Prod. Res., 7,3,233-240.
Artamonov, G.T. [1976], "Productivity of a Two-Instrument Discrete Processing Line in the Presence of Failures", Kibernetika, 3,126-130
(Eng. tr. Cybernetics, 12,3,464-468 (1977)).
(In Russian).
Avi-Itzhak, B. and Yadin, M. [1965], "A Sequence of Two Servers with No
Intermediate Queue", Manag. Sci., 11,5,553-564.
Barlow, R.E., and Proschan, F. [19751, Statistical Theory of Reliability
and Life Testing, Holt, Rinehart, and Winston.
Barten, K. [19621, "A Queueing Simulator for Determining Optimum Inventory
Levels in a Sequential Process", J. Ind. Eng., 13,4,247-252.
Baskett, F., Chandy, K.M., Muntz, R.R., and Palacios, F.G. [1975], "Open,
Closed and Mixed Networks of Queues with Different Classes of
Customers", J. ACM, 22,2,248-260.
Bellman, R.E.
[1970], Introduction to Matrix Analysis, McGraw-Hill.
Berman, O. [1979], "Efficiency and Porduction Rate of a Transfer Line
with Two Machines and a Finite Storage Buffer," Report LIDS-R-899,
Laboratory for Information and Decision Systems, Massachusetts
Institute of Technology
Boyce, W.E. and DiPrima, R.C. [1969], Elementary Differential Equations
and Boundary Value Problems, John Wiley.
Buzacott, J.A. [1967a], "Markov Chain Analysis of Automatic Transfer
Line with Buffer Stock", unpublished Ph.D. Thesis, Department of
Engineering Production, University of Birmingham.
Buzacott, J.A. 11967b], "Automatic Transfer Lines with Buffer Stocks",
Int. J. Prod. Res., 5,3,183-200.
Buzacott, J.A. [1968], "Prediction of the Efficiency of Production
Systems without Internal Storage", Int. J. Prod. Res., 6,3,173-188.
Buzacott, J.A. [1969], "Methods of Reliability Analysis of Production
Systems Subject to Breakdowns", in D. Grouchko Ed., Operations
Research and Reliability, Proc. of a NATO Conf. (Turin, Italy,
June 24 - July 4, 1969), 211-232.
-194-
-195-
Buzacott, J.A. [1972], "The Effect of Station Breakdowns and Random
Processing Times on the Capacity of Flow Lines with In-Process
Storage", AIIE Trans., 4,4,308-312.
Buzacott, J.A. and Hanifin, L.E. [1978], "Models of Automatic Transfer
Lines with Inventory Banks - A Review and Comparison", AIIE Trans.,
10,2,197-207.
Buzacott, J.A. and Shanthikumar, J.G. [19781, "Models for Understanding
Flexible Manufacturing Systems", Department of Industrial Engineering,
University of Toronto, working paper 78-010.
Denning, P.J. and Buzen, J.P. [1977], "Operational Analysis of Queueing
Networks", Preprint of the Third International Symposium of Modelling
and Performance Evaluation of Computer Systems (Bonn, W. Germany,
October 3 - 5, 1977).
Feller, W. [1966], An Introduction to Probability Theory and Its
Applications, Vol. II, John Wiley.
Freeman, M.C. [1964], "The Effects of Breakdowns and Interstage Storage
on Production Line Capacity", J. Ind. Eng., 15,4,194-200.
Gershwin, S.B. [1973a], "Reliability and Storage Size, Part I", unpublished
memorandum No. FM 44700-107A, The Charles Stark Draper Laboratory.
Gershwin, S.B. and Schick, I.C. [1977], "Reliability and Storage Size",
in Athans et.al., "Complex Materials Handling and Assembly Systems,
Second Interim Progress Report", Report ESL-IR-771, Electronic
Systems Laboratory, Massachusetts Institute of Technology.
Gershwin, S.B. and Berman, 0. [1978], "Analysis of Transfer Lines
Consisting of Two Unreliable Stages with Random Processing Times
and Finite Storage Buffers",- "Complex Materials Handling and
Assembly Systems, Volume VII", Report ESL-FR-834-7, Electronic
Systems Laboratory, Massachusetts Institute of Technology.
Golub, G.W. [1969], "Matrix Decomposition and Statistical Calculations"
Technical Report CS 124, Computer Science Department, Stanford
University.
Golub, G.W. and Kahan, W. [1965], "Calculating the Singular Values and
Pseudo-Inverse of a Matrix". J. SIAM Numerical Analysis B, 2,2,
205-224.
Gordon, W.J. and Newell, G.F. [1967a], "Closed Queueing Systems with
Exponential Servers", Op. Res., 15,2,254-265.
-196-
Hadley, G.
[19641,
Linear Algebra, Addison-Wesley.
Hanifin, L.E., Buzacott, J.A. and Taraman, K.S. 11975], "A Comparison of
Analytical and Simulation Models of Transfer Lines", SME Technical
Paper EM75-374.
Hanifin, L.E., Liberty, S.G., and Taraman, K. [1975], "Improved Transfer
Line Efficiency Utilizing Systems Simulation", Technical Paper
MR75-169, Society of Manufacturing Engineers.
Hatcher, J.M. [1969], "The Effect of Internal Storage on the Production
Rate of a Series of Stages Having Exponential Service Times",
AIIE Trans., 1,2,150-156.
Hildebrand, K.K. [1968], "On the Capacity of Tandem Server, Finite
Queue, Service Systems", Op. Res., 16,72-82.
Hillier, F.S. and Boling, R.W. [1966], "The Effect of Some Design Factors
on the Efficiency of ProductiQn Lines with Variable Operation
Times", J. Ind. Eng., 17,12,651-658.
Ho,
Y.-C., Eyler, M.A., and Chien, T.-T. 11979], "A Gradient Technique
for General Buffer Storage Design in a Production Line," Proc.
1978 IEEE Conference on Decision and Control (San Diego, January
10-12, 1979).
Hunt, G,C, [1956], "Sequential Arrays of Waiting Lines", Op. Res., 4,6,
674-683,
Jackson, J.R. 11963],
10,1,131-142.
"Job-Shop-Like Queueing Systems", Manag. Sci.,
Karlin, S. 11968], A First Course in Stochastic Processes, Academic Press.
Kay, E.
[1972], "Buffer Stocks in Automatic Transfer Lines", Int. J.
Prod. Res., 10,2,155-165.
Knott, A,D. [11970a], "The Inefficiency of a Series of Work Stations A Simple Formula", Int. J. Prod. Res., 8,2,109-119.
Knott, A.D. 11970b], "The Effect of Internal Storage on the Production
Rate- of a Series of Stages Having Exponential Service Times",
AIIE Trans. (letters), 2,3,273.
Kruaeer, T.A. and Love, R.F. I1970], "A Model for Optimizing the Buffer
Inventory Storage Size in a Sequential Production System", AIIE
*Trans., 2,1,64-69.
Lam, S.S. 119771],-Queueing Networks with Population Size Constraints",
-IBM J. Res. Dev., 21,4,370-378.
-197-
Masso, J.A. [1973], "Stochastic Lines and Inter-Stage Storages", unpublished
M.S. Thesis, Department of Industrial Engineering, Texas Tech
University.
Masso, J. and Smith, M.L. [1974], "Interstage Storages for Three Stage
Lines Subject to Stochastic Failures", AIIE Trans., 6,4,354-358.
Morse, P.M. [1965], Queues, Inventories and Maintenance, John Wiley.
Muth, E.J. [1973], "The Production Rate of a Series of Work Stations with
Variable Service Times", Int. J. Prod. Res., 11,2,155-169.
Neuts, M.F. [1968], "Two Queues in Series with a Finite, Intermediate
Waitingroom", J. Appl. Probl., 5,123-142.
Neuts, M.F. [1970], "Two Servers in Series, Studied in Terms of a Markov
Renewal Branching Process", Adv. Appl. Prob., 2,110-149.
Okamura, K. and Yamashina, H. [1977], "Analysis of the Effect of Buffer
Storage Capacity in Transfer Line Systems", AIIE Trans., 9,2,127-135.
Patterson, R.L. [1964], "Markov Processes Occurring in the Theory of
Traffic Flow Through an N-Stage Stochastic Service System", J.
Ind. Eng., 15,4,188-193.
Pomerance, B. [1979], "Analysis of a Three-Machine Tranfer Line", unpublished
B.S. Thesis, Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology.
Rao, N.P. [1975a], "On the Mean Production Rate of a Two-Stage Production
J., Prod. Res., 13,2,207-217.
System of the Tandem Type", Int.
Rao, N.P. [1975b], "Two-Stage Production Systems with Intermediate Storage",
AIIE Trans., 7,4,414-421.
Schick, I.C. and Gershwin, S.B. [19781, "Modelling and Analysis of Unreliable Transfer Lines with Finite Interstage Buffers", "Complex
Materials Handling and Assembly Systems, Volume VI", Report ESLFR-834-6, Electronic Systems Laboratory, Massachusetts Institute
of Technology.
Schweitzer, P.J. [1977], "Maximum Throughput in Finite-Capacity Open
Queueing Networks with Product-Form Solutions", Manag. Sci.,
24, 217-223.
Sevast'yanov, B.A. [19621, "Influence of Storage Bin Capacity on the Average
Standstill Time of a Production Line", Teoriya Veroyatnostey i
(Eng. tr. Theory of
ee Primeneniya, 7,4,438-447 In Russian).
Probability and Its Aplications, 7,4,429-438 (1962)).
Sheskin, T.J. [1974], "Allocation of Interstage Storage Along an Automatic
Transfer Production Line with Discrete Flow', unpublished Ph.D.
Thesis, Department of Industrial and Management Systems Engineering,
Pennsylvania State University.
-198-
Sheskin, T.J. 11976], "Allocation of Interstage Storage Along an Automatic
Production Line', AIIE Trans., 8,1,146-152.
Solberg, J.J. 11977], "A Mathematical Model of Computerized Manufacturing
Systems", Conference on Optimal Planning of Computerized Manufacturing
Systems Project (Purdue University, November 21 - 22, 1977).
Suzuki, T. [1964], "On a Tandem Queue with Blocking", J. Op. Res. Soc.
Japan, 6,137-257.
Tobey, R., et al. 11969], "PL/I - FORMAC Symbolic Mathematics Interpreter",
IBM Contributed Program Library, 360D-03.3.004.
Trufyn, N. [n.d.], "Guide to FORMAC", Institute of Computer Science,
University of Toronto.
Vladzievskii, A.P. [1952], "Probabilistic Law of Operation and Internal
Storage of Automatic Lines", Avtomatika i Telemekhanika, 13,3,227-281
(In Russian).